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Theorem wemaplem2 8615
Description: Lemma for wemapso 8619. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Hypotheses
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
wemaplem2.a (𝜑𝐴 ∈ V)
wemaplem2.p (𝜑𝑃 ∈ (𝐵𝑚 𝐴))
wemaplem2.x (𝜑𝑋 ∈ (𝐵𝑚 𝐴))
wemaplem2.q (𝜑𝑄 ∈ (𝐵𝑚 𝐴))
wemaplem2.r (𝜑𝑅 Or 𝐴)
wemaplem2.s (𝜑𝑆 Po 𝐵)
wemaplem2.px1 (𝜑𝑎𝐴)
wemaplem2.px2 (𝜑 → (𝑃𝑎)𝑆(𝑋𝑎))
wemaplem2.px3 (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))
wemaplem2.xq1 (𝜑𝑏𝐴)
wemaplem2.xq2 (𝜑 → (𝑋𝑏)𝑆(𝑄𝑏))
wemaplem2.xq3 (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))
Assertion
Ref Expression
wemaplem2 (𝜑𝑃𝑇𝑄)
Distinct variable groups:   𝑎,𝑏,𝑐,𝑥,𝐵   𝑇,𝑎,𝑏,𝑐   𝑤,𝑎,𝑦,𝑧,𝑋,𝑏,𝑐,𝑥   𝐴,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑃,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑄,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑅,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧   𝑆,𝑎,𝑏,𝑐,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏,𝑐)   𝐵(𝑦,𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wemaplem2
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 wemaplem2.px1 . . . 4 (𝜑𝑎𝐴)
2 wemaplem2.xq1 . . . 4 (𝜑𝑏𝐴)
31, 2ifcld 4273 . . 3 (𝜑 → if(𝑎𝑅𝑏, 𝑎, 𝑏) ∈ 𝐴)
4 wemaplem2.px2 . . . . . . 7 (𝜑 → (𝑃𝑎)𝑆(𝑋𝑎))
54adantr 472 . . . . . 6 ((𝜑𝑎𝑅𝑏) → (𝑃𝑎)𝑆(𝑋𝑎))
6 wemaplem2.xq3 . . . . . . . 8 (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))
7 breq1 4805 . . . . . . . . . 10 (𝑐 = 𝑎 → (𝑐𝑅𝑏𝑎𝑅𝑏))
8 fveq2 6350 . . . . . . . . . . 11 (𝑐 = 𝑎 → (𝑋𝑐) = (𝑋𝑎))
9 fveq2 6350 . . . . . . . . . . 11 (𝑐 = 𝑎 → (𝑄𝑐) = (𝑄𝑎))
108, 9eqeq12d 2773 . . . . . . . . . 10 (𝑐 = 𝑎 → ((𝑋𝑐) = (𝑄𝑐) ↔ (𝑋𝑎) = (𝑄𝑎)))
117, 10imbi12d 333 . . . . . . . . 9 (𝑐 = 𝑎 → ((𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)) ↔ (𝑎𝑅𝑏 → (𝑋𝑎) = (𝑄𝑎))))
1211rspcva 3445 . . . . . . . 8 ((𝑎𝐴 ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → (𝑎𝑅𝑏 → (𝑋𝑎) = (𝑄𝑎)))
131, 6, 12syl2anc 696 . . . . . . 7 (𝜑 → (𝑎𝑅𝑏 → (𝑋𝑎) = (𝑄𝑎)))
1413imp 444 . . . . . 6 ((𝜑𝑎𝑅𝑏) → (𝑋𝑎) = (𝑄𝑎))
155, 14breqtrd 4828 . . . . 5 ((𝜑𝑎𝑅𝑏) → (𝑃𝑎)𝑆(𝑄𝑎))
16 iftrue 4234 . . . . . . . 8 (𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑎)
1716fveq2d 6354 . . . . . . 7 (𝑎𝑅𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃𝑎))
1816fveq2d 6354 . . . . . . 7 (𝑎𝑅𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄𝑎))
1917, 18breq12d 4815 . . . . . 6 (𝑎𝑅𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑎)𝑆(𝑄𝑎)))
2019adantl 473 . . . . 5 ((𝜑𝑎𝑅𝑏) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑎)𝑆(𝑄𝑎)))
2115, 20mpbird 247 . . . 4 ((𝜑𝑎𝑅𝑏) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
22 wemaplem2.s . . . . . . 7 (𝜑𝑆 Po 𝐵)
2322adantr 472 . . . . . 6 ((𝜑𝑎 = 𝑏) → 𝑆 Po 𝐵)
24 wemaplem2.p . . . . . . . . . 10 (𝜑𝑃 ∈ (𝐵𝑚 𝐴))
25 elmapi 8043 . . . . . . . . . 10 (𝑃 ∈ (𝐵𝑚 𝐴) → 𝑃:𝐴𝐵)
2624, 25syl 17 . . . . . . . . 9 (𝜑𝑃:𝐴𝐵)
2726, 2ffvelrnd 6521 . . . . . . . 8 (𝜑 → (𝑃𝑏) ∈ 𝐵)
28 wemaplem2.x . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐵𝑚 𝐴))
29 elmapi 8043 . . . . . . . . . 10 (𝑋 ∈ (𝐵𝑚 𝐴) → 𝑋:𝐴𝐵)
3028, 29syl 17 . . . . . . . . 9 (𝜑𝑋:𝐴𝐵)
3130, 2ffvelrnd 6521 . . . . . . . 8 (𝜑 → (𝑋𝑏) ∈ 𝐵)
32 wemaplem2.q . . . . . . . . . 10 (𝜑𝑄 ∈ (𝐵𝑚 𝐴))
33 elmapi 8043 . . . . . . . . . 10 (𝑄 ∈ (𝐵𝑚 𝐴) → 𝑄:𝐴𝐵)
3432, 33syl 17 . . . . . . . . 9 (𝜑𝑄:𝐴𝐵)
3534, 2ffvelrnd 6521 . . . . . . . 8 (𝜑 → (𝑄𝑏) ∈ 𝐵)
3627, 31, 353jca 1123 . . . . . . 7 (𝜑 → ((𝑃𝑏) ∈ 𝐵 ∧ (𝑋𝑏) ∈ 𝐵 ∧ (𝑄𝑏) ∈ 𝐵))
3736adantr 472 . . . . . 6 ((𝜑𝑎 = 𝑏) → ((𝑃𝑏) ∈ 𝐵 ∧ (𝑋𝑏) ∈ 𝐵 ∧ (𝑄𝑏) ∈ 𝐵))
38 fveq2 6350 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑃𝑎) = (𝑃𝑏))
39 fveq2 6350 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑋𝑎) = (𝑋𝑏))
4038, 39breq12d 4815 . . . . . . . 8 (𝑎 = 𝑏 → ((𝑃𝑎)𝑆(𝑋𝑎) ↔ (𝑃𝑏)𝑆(𝑋𝑏)))
414, 40syl5ibcom 235 . . . . . . 7 (𝜑 → (𝑎 = 𝑏 → (𝑃𝑏)𝑆(𝑋𝑏)))
4241imp 444 . . . . . 6 ((𝜑𝑎 = 𝑏) → (𝑃𝑏)𝑆(𝑋𝑏))
43 wemaplem2.xq2 . . . . . . 7 (𝜑 → (𝑋𝑏)𝑆(𝑄𝑏))
4443adantr 472 . . . . . 6 ((𝜑𝑎 = 𝑏) → (𝑋𝑏)𝑆(𝑄𝑏))
45 potr 5197 . . . . . . 7 ((𝑆 Po 𝐵 ∧ ((𝑃𝑏) ∈ 𝐵 ∧ (𝑋𝑏) ∈ 𝐵 ∧ (𝑄𝑏) ∈ 𝐵)) → (((𝑃𝑏)𝑆(𝑋𝑏) ∧ (𝑋𝑏)𝑆(𝑄𝑏)) → (𝑃𝑏)𝑆(𝑄𝑏)))
4645imp 444 . . . . . 6 (((𝑆 Po 𝐵 ∧ ((𝑃𝑏) ∈ 𝐵 ∧ (𝑋𝑏) ∈ 𝐵 ∧ (𝑄𝑏) ∈ 𝐵)) ∧ ((𝑃𝑏)𝑆(𝑋𝑏) ∧ (𝑋𝑏)𝑆(𝑄𝑏))) → (𝑃𝑏)𝑆(𝑄𝑏))
4723, 37, 42, 44, 46syl22anc 1478 . . . . 5 ((𝜑𝑎 = 𝑏) → (𝑃𝑏)𝑆(𝑄𝑏))
48 ifeq1 4232 . . . . . . . . 9 (𝑎 = 𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = if(𝑎𝑅𝑏, 𝑏, 𝑏))
49 ifid 4267 . . . . . . . . 9 if(𝑎𝑅𝑏, 𝑏, 𝑏) = 𝑏
5048, 49syl6eq 2808 . . . . . . . 8 (𝑎 = 𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
5150fveq2d 6354 . . . . . . 7 (𝑎 = 𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃𝑏))
5250fveq2d 6354 . . . . . . 7 (𝑎 = 𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄𝑏))
5351, 52breq12d 4815 . . . . . 6 (𝑎 = 𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑏)𝑆(𝑄𝑏)))
5453adantl 473 . . . . 5 ((𝜑𝑎 = 𝑏) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑏)𝑆(𝑄𝑏)))
5547, 54mpbird 247 . . . 4 ((𝜑𝑎 = 𝑏) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
56 wemaplem2.px3 . . . . . . . 8 (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))
57 breq1 4805 . . . . . . . . . 10 (𝑐 = 𝑏 → (𝑐𝑅𝑎𝑏𝑅𝑎))
58 fveq2 6350 . . . . . . . . . . 11 (𝑐 = 𝑏 → (𝑃𝑐) = (𝑃𝑏))
59 fveq2 6350 . . . . . . . . . . 11 (𝑐 = 𝑏 → (𝑋𝑐) = (𝑋𝑏))
6058, 59eqeq12d 2773 . . . . . . . . . 10 (𝑐 = 𝑏 → ((𝑃𝑐) = (𝑋𝑐) ↔ (𝑃𝑏) = (𝑋𝑏)))
6157, 60imbi12d 333 . . . . . . . . 9 (𝑐 = 𝑏 → ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ↔ (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑋𝑏))))
6261rspcva 3445 . . . . . . . 8 ((𝑏𝐴 ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))) → (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑋𝑏)))
632, 56, 62syl2anc 696 . . . . . . 7 (𝜑 → (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑋𝑏)))
6463imp 444 . . . . . 6 ((𝜑𝑏𝑅𝑎) → (𝑃𝑏) = (𝑋𝑏))
6543adantr 472 . . . . . 6 ((𝜑𝑏𝑅𝑎) → (𝑋𝑏)𝑆(𝑄𝑏))
6664, 65eqbrtrd 4824 . . . . 5 ((𝜑𝑏𝑅𝑎) → (𝑃𝑏)𝑆(𝑄𝑏))
67 wemaplem2.r . . . . . . . . 9 (𝜑𝑅 Or 𝐴)
68 sopo 5202 . . . . . . . . 9 (𝑅 Or 𝐴𝑅 Po 𝐴)
6967, 68syl 17 . . . . . . . 8 (𝜑𝑅 Po 𝐴)
70 po2nr 5198 . . . . . . . 8 ((𝑅 Po 𝐴 ∧ (𝑏𝐴𝑎𝐴)) → ¬ (𝑏𝑅𝑎𝑎𝑅𝑏))
7169, 2, 1, 70syl12anc 1475 . . . . . . 7 (𝜑 → ¬ (𝑏𝑅𝑎𝑎𝑅𝑏))
72 nan 605 . . . . . . 7 ((𝜑 → ¬ (𝑏𝑅𝑎𝑎𝑅𝑏)) ↔ ((𝜑𝑏𝑅𝑎) → ¬ 𝑎𝑅𝑏))
7371, 72mpbi 220 . . . . . 6 ((𝜑𝑏𝑅𝑎) → ¬ 𝑎𝑅𝑏)
74 iffalse 4237 . . . . . . . 8 𝑎𝑅𝑏 → if(𝑎𝑅𝑏, 𝑎, 𝑏) = 𝑏)
7574fveq2d 6354 . . . . . . 7 𝑎𝑅𝑏 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑃𝑏))
7674fveq2d 6354 . . . . . . 7 𝑎𝑅𝑏 → (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) = (𝑄𝑏))
7775, 76breq12d 4815 . . . . . 6 𝑎𝑅𝑏 → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑏)𝑆(𝑄𝑏)))
7873, 77syl 17 . . . . 5 ((𝜑𝑏𝑅𝑎) → ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ↔ (𝑃𝑏)𝑆(𝑄𝑏)))
7966, 78mpbird 247 . . . 4 ((𝜑𝑏𝑅𝑎) → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
80 solin 5208 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
8167, 1, 2, 80syl12anc 1475 . . . 4 (𝜑 → (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
8221, 55, 79, 81mpjao3dan 1542 . . 3 (𝜑 → (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
83 r19.26 3200 . . . . 5 (∀𝑐𝐴 ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) ↔ (∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))
8456, 6, 83sylanbrc 701 . . . 4 (𝜑 → ∀𝑐𝐴 ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))
8567, 1, 23jca 1123 . . . . 5 (𝜑 → (𝑅 Or 𝐴𝑎𝐴𝑏𝐴))
86 prth 596 . . . . . . 7 (((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → ((𝑐𝑅𝑎𝑐𝑅𝑏) → ((𝑃𝑐) = (𝑋𝑐) ∧ (𝑋𝑐) = (𝑄𝑐))))
87 eqtr 2777 . . . . . . 7 (((𝑃𝑐) = (𝑋𝑐) ∧ (𝑋𝑐) = (𝑄𝑐)) → (𝑃𝑐) = (𝑄𝑐))
8886, 87syl6 35 . . . . . 6 (((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → ((𝑐𝑅𝑎𝑐𝑅𝑏) → (𝑃𝑐) = (𝑄𝑐)))
8988ralimi 3088 . . . . 5 (∀𝑐𝐴 ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → ∀𝑐𝐴 ((𝑐𝑅𝑎𝑐𝑅𝑏) → (𝑃𝑐) = (𝑄𝑐)))
90 simpl1 1228 . . . . . . . . 9 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → 𝑅 Or 𝐴)
91 simpr 479 . . . . . . . . 9 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → 𝑐𝐴)
92 simpl2 1230 . . . . . . . . 9 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → 𝑎𝐴)
93 simpl3 1232 . . . . . . . . 9 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → 𝑏𝐴)
94 soltmin 5688 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝑐𝐴𝑎𝐴𝑏𝐴)) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) ↔ (𝑐𝑅𝑎𝑐𝑅𝑏)))
9590, 91, 92, 93, 94syl13anc 1479 . . . . . . . 8 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) ↔ (𝑐𝑅𝑎𝑐𝑅𝑏)))
9695biimpd 219 . . . . . . 7 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑐𝑅𝑎𝑐𝑅𝑏)))
9796imim1d 82 . . . . . 6 (((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴) → (((𝑐𝑅𝑎𝑐𝑅𝑏) → (𝑃𝑐) = (𝑄𝑐)) → (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
9897ralimdva 3098 . . . . 5 ((𝑅 Or 𝐴𝑎𝐴𝑏𝐴) → (∀𝑐𝐴 ((𝑐𝑅𝑎𝑐𝑅𝑏) → (𝑃𝑐) = (𝑄𝑐)) → ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
9985, 89, 98syl2im 40 . . . 4 (𝜑 → (∀𝑐𝐴 ((𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)) ∧ (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
10084, 99mpd 15 . . 3 (𝜑 → ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐)))
101 fveq2 6350 . . . . . 6 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑑) = (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
102 fveq2 6350 . . . . . 6 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑄𝑑) = (𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)))
103101, 102breq12d 4815 . . . . 5 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → ((𝑃𝑑)𝑆(𝑄𝑑) ↔ (𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏))))
104 breq2 4806 . . . . . . 7 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑐𝑅𝑑𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏)))
105104imbi1d 330 . . . . . 6 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → ((𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐)) ↔ (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
106105ralbidv 3122 . . . . 5 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐)) ↔ ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐))))
107103, 106anbi12d 749 . . . 4 (𝑑 = if(𝑎𝑅𝑏, 𝑎, 𝑏) → (((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐))) ↔ ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ∧ ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐)))))
108107rspcev 3447 . . 3 ((if(𝑎𝑅𝑏, 𝑎, 𝑏) ∈ 𝐴 ∧ ((𝑃‘if(𝑎𝑅𝑏, 𝑎, 𝑏))𝑆(𝑄‘if(𝑎𝑅𝑏, 𝑎, 𝑏)) ∧ ∀𝑐𝐴 (𝑐𝑅if(𝑎𝑅𝑏, 𝑎, 𝑏) → (𝑃𝑐) = (𝑄𝑐)))) → ∃𝑑𝐴 ((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐))))
1093, 82, 100, 108syl12anc 1475 . 2 (𝜑 → ∃𝑑𝐴 ((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐))))
110 wemapso.t . . . 4 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
111110wemaplem1 8614 . . 3 ((𝑃 ∈ (𝐵𝑚 𝐴) ∧ 𝑄 ∈ (𝐵𝑚 𝐴)) → (𝑃𝑇𝑄 ↔ ∃𝑑𝐴 ((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐)))))
11224, 32, 111syl2anc 696 . 2 (𝜑 → (𝑃𝑇𝑄 ↔ ∃𝑑𝐴 ((𝑃𝑑)𝑆(𝑄𝑑) ∧ ∀𝑐𝐴 (𝑐𝑅𝑑 → (𝑃𝑐) = (𝑄𝑐)))))
113109, 112mpbird 247 1 (𝜑𝑃𝑇𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3o 1071  w3a 1072   = wceq 1630  wcel 2137  wral 3048  wrex 3049  Vcvv 3338  ifcif 4228   class class class wbr 4802  {copab 4862   Po wpo 5183   Or wor 5184  wf 6043  cfv 6047  (class class class)co 6811  𝑚 cmap 8021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-po 5185  df-so 5186  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-1st 7331  df-2nd 7332  df-map 8023
This theorem is referenced by:  wemaplem3  8616
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