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Theorem f1resrcmplf1dlem 35079
Description: Lemma for f1resrcmplf1d 35080. (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 6738 . . . . . 6 (𝜑𝐹 Fn 𝐴)
4 fnfvima 7253 . . . . . 6 ((𝐹 Fn 𝐴𝐶𝐴𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
53, 4syl3an1 1162 . . . . 5 ((𝜑𝐶𝐴𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
61, 5syl3an2 1163 . . . 4 ((𝜑𝜑𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
763anidm12 1418 . . 3 ((𝜑𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
87ex 412 . 2 (𝜑 → (𝑋𝐶 → (𝐹𝑋) ∈ (𝐹𝐶)))
9 f1resrcmplf1dlem.2 . . . . 5 (𝜑𝐷𝐴)
10 fnfvima 7253 . . . . . 6 ((𝐹 Fn 𝐴𝐷𝐴𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
113, 10syl3an1 1162 . . . . 5 ((𝜑𝐷𝐴𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
129, 11syl3an2 1163 . . . 4 ((𝜑𝜑𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
13123anidm12 1418 . . 3 ((𝜑𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
1413ex 412 . 2 (𝜑 → (𝑌𝐷 → (𝐹𝑌) ∈ (𝐹𝐷)))
15 f1resrcmplf1dlem.4 . . . . 5 (𝜑 → ((𝐹𝐶) ∩ (𝐹𝐷)) = ∅)
16 disjne 4461 . . . . 5 ((((𝐹𝐶) ∩ (𝐹𝐷)) = ∅ ∧ (𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌))
1715, 16syl3an1 1162 . . . 4 ((𝜑 ∧ (𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌))
18173expib 1121 . . 3 (𝜑 → (((𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌)))
19 neneq 2944 . . . 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 395   = wceq 1537  wcel 2106  wne 2938  cin 3962  wss 3963  c0 4339  cima 5692   Fn wfn 6558  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  f1resrcmplf1d  35080
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