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Theorem f1ocpbllem 16499
Description: Lemma for f1ocpbl 16500. (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 6355 . . . . 5 (𝐹:𝑉1-1-onto𝑋𝐹:𝑉1-1𝑋)
31, 2syl 17 . . . 4 (𝜑𝐹:𝑉1-1𝑋)
433ad2ant1 1164 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐹:𝑉1-1𝑋)
5 simp2l 1257 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐴𝑉)
6 simp3l 1259 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐶𝑉)
7 f1fveq 6747 . . 3 ((𝐹:𝑉1-1𝑋 ∧ (𝐴𝑉𝐶𝑉)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
84, 5, 6, 7syl12anc 866 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹𝐴) = (𝐹𝐶) ↔ 𝐴 = 𝐶))
9 simp2r 1258 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐵𝑉)
10 simp3r 1260 . . 3 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → 𝐷𝑉)
11 f1fveq 6747 . . 3 ((𝐹:𝑉1-1𝑋 ∧ (𝐵𝑉𝐷𝑉)) → ((𝐹𝐵) = (𝐹𝐷) ↔ 𝐵 = 𝐷))
124, 9, 10, 11syl12anc 866 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐹𝐵) = (𝐹𝐷) ↔ 𝐵 = 𝐷))
138, 12anbi12d 625 1 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  1-1wf1 6098  1-1-ontowf1o 6100  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-f1o 6108  df-fv 6109
This theorem is referenced by:  f1ocpbl  16500  f1olecpbl  16502
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