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Theorem f1ocpbllem 17505
Description: Lemma for f1ocpbl 17506. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypothesis
Ref Expression
f1ocpbl.f (𝜑𝐹:𝑉1-1-onto𝑋)
Assertion
Ref Expression
f1ocpbllem ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem f1ocpbllem
StepHypRef Expression
1 f1ocpbl.f . . . . 5 (𝜑𝐹:𝑉1-1-onto𝑋)
2 f1of1 6833 . . . . 5 (𝐹:𝑉1-1-onto𝑋𝐹:𝑉1-1𝑋)
31, 2syl 17 . . . 4 (𝜑𝐹:𝑉1-1𝑋)
433ad2ant1 1130 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐹:𝑉1-1𝑋)
5 simp2l 1196 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐴𝑉)
6 simp3l 1198 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐶𝑉)
7 f1fveq 7268 . . 3 ((𝐹:𝑉1-1𝑋 ∧ (𝐴𝑉𝐶𝑉)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
84, 5, 6, 7syl12anc 835 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
9 simp2r 1197 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐵𝑉)
10 simp3r 1199 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐷𝑉)
11 f1fveq 7268 . . 3 ((𝐹:𝑉1-1𝑋 ∧ (𝐵𝑉𝐷𝑉)) → ((𝐹𝐵) = (𝐹𝐷) ↔ 𝐵 = 𝐷))
124, 9, 10, 11syl12anc 835 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹𝐵) = (𝐹𝐷) ↔ 𝐵 = 𝐷))
138, 12anbi12d 630 1 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  1-1wf1 6540  1-1-ontowf1o 6542  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-f1o 6550  df-fv 6551
This theorem is referenced by:  f1ocpbl  17506  f1olecpbl  17508
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