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Theorem f1resrcmplf1dlem 32469
Description: Lemma for f1resrcmplf1d 32470. (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 6488 . . . . . 6 (𝜑𝐹 Fn 𝐴)
4 fnfvima 6973 . . . . . 6 ((𝐹 Fn 𝐴𝐶𝐴𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
53, 4syl3an1 1160 . . . . 5 ((𝜑𝐶𝐴𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
61, 5syl3an2 1161 . . . 4 ((𝜑𝜑𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
763anidm12 1416 . . 3 ((𝜑𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
87ex 416 . 2 (𝜑 → (𝑋𝐶 → (𝐹𝑋) ∈ (𝐹𝐶)))
9 f1resrcmplf1dlem.2 . . . . 5 (𝜑𝐷𝐴)
10 fnfvima 6973 . . . . . 6 ((𝐹 Fn 𝐴𝐷𝐴𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
113, 10syl3an1 1160 . . . . 5 ((𝜑𝐷𝐴𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
129, 11syl3an2 1161 . . . 4 ((𝜑𝜑𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
13123anidm12 1416 . . 3 ((𝜑𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
1413ex 416 . 2 (𝜑 → (𝑌𝐷 → (𝐹𝑌) ∈ (𝐹𝐷)))
15 f1resrcmplf1dlem.4 . . . . 5 (𝜑 → ((𝐹𝐶) ∩ (𝐹𝐷)) = ∅)
16 disjne 4362 . . . . 5 ((((𝐹𝐶) ∩ (𝐹𝐷)) = ∅ ∧ (𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌))
1715, 16syl3an1 1160 . . . 4 ((𝜑 ∧ (𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌))
18173expib 1119 . . 3 (𝜑 → (((𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌)))
19 neneq 2993 . . . 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 610 1 (𝜑 → ((𝑋𝐶𝑌𝐷) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  wne 2987  cin 3880  wss 3881  c0 4243  cima 5522   Fn wfn 6319  wf 6320  cfv 6324
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332
This theorem is referenced by:  f1resrcmplf1d  32470
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