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Theorem f1resrcmplf1dlem 33681
Description: Lemma for f1resrcmplf1d 33682. (Contributed by BTernaryTau, 27-Sep-2023.)
Hypotheses
Ref Expression
f1resrcmplf1dlem.1 (𝜑𝐶𝐴)
f1resrcmplf1dlem.2 (𝜑𝐷𝐴)
f1resrcmplf1dlem.3 (𝜑𝐹:𝐴𝐵)
f1resrcmplf1dlem.4 (𝜑 → ((𝐹𝐶) ∩ (𝐹𝐷)) = ∅)
Assertion
Ref Expression
f1resrcmplf1dlem (𝜑 → ((𝑋𝐶𝑌𝐷) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))

Proof of Theorem f1resrcmplf1dlem
StepHypRef Expression
1 f1resrcmplf1dlem.1 . . . . 5 (𝜑𝐶𝐴)
2 f1resrcmplf1dlem.3 . . . . . . 7 (𝜑𝐹:𝐴𝐵)
32ffnd 6669 . . . . . 6 (𝜑𝐹 Fn 𝐴)
4 fnfvima 7182 . . . . . 6 ((𝐹 Fn 𝐴𝐶𝐴𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
53, 4syl3an1 1163 . . . . 5 ((𝜑𝐶𝐴𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
61, 5syl3an2 1164 . . . 4 ((𝜑𝜑𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
763anidm12 1419 . . 3 ((𝜑𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
87ex 413 . 2 (𝜑 → (𝑋𝐶 → (𝐹𝑋) ∈ (𝐹𝐶)))
9 f1resrcmplf1dlem.2 . . . . 5 (𝜑𝐷𝐴)
10 fnfvima 7182 . . . . . 6 ((𝐹 Fn 𝐴𝐷𝐴𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
113, 10syl3an1 1163 . . . . 5 ((𝜑𝐷𝐴𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
129, 11syl3an2 1164 . . . 4 ((𝜑𝜑𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
13123anidm12 1419 . . 3 ((𝜑𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
1413ex 413 . 2 (𝜑 → (𝑌𝐷 → (𝐹𝑌) ∈ (𝐹𝐷)))
15 f1resrcmplf1dlem.4 . . . . 5 (𝜑 → ((𝐹𝐶) ∩ (𝐹𝐷)) = ∅)
16 disjne 4414 . . . . 5 ((((𝐹𝐶) ∩ (𝐹𝐷)) = ∅ ∧ (𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌))
1715, 16syl3an1 1163 . . . 4 ((𝜑 ∧ (𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌))
18173expib 1122 . . 3 (𝜑 → (((𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌)))
19 neneq 2949 . . . 4 ((𝐹𝑋) ≠ (𝐹𝑌) → ¬ (𝐹𝑋) = (𝐹𝑌))
2019pm2.21d 121 . . 3 ((𝐹𝑋) ≠ (𝐹𝑌) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌))
2118, 20syl6 35 . 2 (𝜑 → (((𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
228, 14, 21syl2and 608 1 (𝜑 → ((𝑋𝐶𝑌𝐷) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  cin 3909  wss 3910  c0 4282  cima 5636   Fn wfn 6491  wf 6492  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504
This theorem is referenced by:  f1resrcmplf1d  33682
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