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Theorem tfrlem5 7633
Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem5 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,,𝑢,𝑣,𝐹   𝐴,𝑔,
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑢,𝑓)

Proof of Theorem tfrlem5
Dummy variables 𝑧 𝑎 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
2 vex 3331 . . 3 𝑔 ∈ V
31, 2tfrlem3a 7630 . 2 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))))
4 vex 3331 . . 3 ∈ V
51, 4tfrlem3a 7630 . 2 (𝐴 ↔ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎))))
6 reeanv 3233 . . 3 (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ↔ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))))
7 simp2ll 1283 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑔 Fn 𝑧)
8 simp3l 1220 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑔𝑢)
9 fnbr 6142 . . . . . . . . 9 ((𝑔 Fn 𝑧𝑥𝑔𝑢) → 𝑥𝑧)
107, 8, 9syl2anc 696 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑧)
11 simp2rl 1285 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fn 𝑤)
12 simp3r 1221 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑣)
13 fnbr 6142 . . . . . . . . 9 (( Fn 𝑤𝑥𝑣) → 𝑥𝑤)
1411, 12, 13syl2anc 696 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑤)
1510, 14elind 3929 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥 ∈ (𝑧𝑤))
16 onin 5903 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤) ∈ On)
17163ad2ant1 1125 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ∈ On)
18 fnfun 6137 . . . . . . . . . 10 (𝑔 Fn 𝑧 → Fun 𝑔)
197, 18syl 17 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fun 𝑔)
20 inss1 3964 . . . . . . . . . 10 (𝑧𝑤) ⊆ 𝑧
21 fndm 6139 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
227, 21syl 17 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → dom 𝑔 = 𝑧)
2320, 22syl5sseqr 3783 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ⊆ dom 𝑔)
2419, 23jca 555 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (Fun 𝑔 ∧ (𝑧𝑤) ⊆ dom 𝑔))
25 fnfun 6137 . . . . . . . . . 10 ( Fn 𝑤 → Fun )
2611, 25syl 17 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fun )
27 inss2 3965 . . . . . . . . . 10 (𝑧𝑤) ⊆ 𝑤
28 fndm 6139 . . . . . . . . . . 11 ( Fn 𝑤 → dom = 𝑤)
2911, 28syl 17 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → dom = 𝑤)
3027, 29syl5sseqr 3783 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ⊆ dom )
3126, 30jca 555 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (Fun ∧ (𝑧𝑤) ⊆ dom ))
32 simp2lr 1284 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎)))
33 ssralv 3795 . . . . . . . . 9 ((𝑧𝑤) ⊆ 𝑧 → (∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝐹‘(𝑔𝑎))))
3420, 32, 33mpsyl 68 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝐹‘(𝑔𝑎)))
35 simp2rr 1286 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))
36 ssralv 3795 . . . . . . . . 9 ((𝑧𝑤) ⊆ 𝑤 → (∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)) → ∀𝑎 ∈ (𝑧𝑤)(𝑎) = (𝐹‘(𝑎))))
3727, 35, 36mpsyl 68 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑎) = (𝐹‘(𝑎)))
3817, 24, 31, 34, 37tfrlem1 7629 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝑎))
39 fveq2 6340 . . . . . . . . 9 (𝑎 = 𝑥 → (𝑔𝑎) = (𝑔𝑥))
40 fveq2 6340 . . . . . . . . 9 (𝑎 = 𝑥 → (𝑎) = (𝑥))
4139, 40eqeq12d 2763 . . . . . . . 8 (𝑎 = 𝑥 → ((𝑔𝑎) = (𝑎) ↔ (𝑔𝑥) = (𝑥)))
4241rspcv 3433 . . . . . . 7 (𝑥 ∈ (𝑧𝑤) → (∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝑎) → (𝑔𝑥) = (𝑥)))
4315, 38, 42sylc 65 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑔𝑥) = (𝑥))
44 funbrfv 6383 . . . . . . 7 (Fun 𝑔 → (𝑥𝑔𝑢 → (𝑔𝑥) = 𝑢))
4519, 8, 44sylc 65 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑔𝑥) = 𝑢)
46 funbrfv 6383 . . . . . . 7 (Fun → (𝑥𝑣 → (𝑥) = 𝑣))
4726, 12, 46sylc 65 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑥) = 𝑣)
4843, 45, 473eqtr3d 2790 . . . . 5 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑢 = 𝑣)
49483exp 1112 . . . 4 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣)))
5049rexlimivv 3162 . . 3 (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
516, 50sylbir 225 . 2 ((∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
523, 5, 51syl2anb 497 1 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1620  wcel 2127  {cab 2734  wral 3038  wrex 3039  cin 3702  wss 3703   class class class wbr 4792  dom cdm 5254  cres 5256  Oncon0 5872  Fun wfun 6031   Fn wfn 6032  cfv 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-ord 5875  df-on 5876  df-iota 6000  df-fun 6039  df-fn 6040  df-fv 6045
This theorem is referenced by:  tfrlem7  7636
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