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Theorem fnwe2val 38038
Description: Lemma for fnwe2 38042. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
fnwe2.su (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
fnwe2.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
Assertion
Ref Expression
fnwe2val (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
Distinct variable groups:   𝑦,𝑈,𝑧,𝑎,𝑏   𝑥,𝑆,𝑦,𝑎,𝑏   𝑥,𝑅,𝑦,𝑎,𝑏   𝑥,𝑧,𝐹,𝑦,𝑎,𝑏   𝑇,𝑎,𝑏
Allowed substitution hints:   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥,𝑦,𝑧)   𝑈(𝑥)

Proof of Theorem fnwe2val
StepHypRef Expression
1 vex 3307 . 2 𝑎 ∈ V
2 vex 3307 . 2 𝑏 ∈ V
3 fveq2 6304 . . . 4 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
4 fveq2 6304 . . . 4 (𝑦 = 𝑏 → (𝐹𝑦) = (𝐹𝑏))
53, 4breqan12d 4776 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥)𝑅(𝐹𝑦) ↔ (𝐹𝑎)𝑅(𝐹𝑏)))
63, 4eqeqan12d 2740 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑎) = (𝐹𝑏)))
7 simpl 474 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑥 = 𝑎)
8 fvex 6314 . . . . . . . 8 (𝐹𝑥) ∈ V
9 fnwe2.su . . . . . . . 8 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
108, 9csbie 3665 . . . . . . 7 (𝐹𝑥) / 𝑧𝑆 = 𝑈
113csbeq1d 3646 . . . . . . 7 (𝑥 = 𝑎(𝐹𝑥) / 𝑧𝑆 = (𝐹𝑎) / 𝑧𝑆)
1210, 11syl5eqr 2772 . . . . . 6 (𝑥 = 𝑎𝑈 = (𝐹𝑎) / 𝑧𝑆)
1312adantr 472 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑈 = (𝐹𝑎) / 𝑧𝑆)
14 simpr 479 . . . . 5 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝑦 = 𝑏)
157, 13, 14breq123d 4774 . . . 4 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑥𝑈𝑦𝑎(𝐹𝑎) / 𝑧𝑆𝑏))
166, 15anbi12d 749 . . 3 ((𝑥 = 𝑎𝑦 = 𝑏) → (((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦) ↔ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
175, 16orbi12d 748 . 2 ((𝑥 = 𝑎𝑦 = 𝑏) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦)) ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏))))
18 fnwe2.t . 2 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
191, 2, 17, 18braba 5096 1 (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1596  csb 3639   class class class wbr 4760  {copab 4820  cfv 6001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-iota 5964  df-fv 6009
This theorem is referenced by:  fnwe2lem2  38040  fnwe2lem3  38041
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