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Theorem f1resrcmplf1dlem 35100
Description: Lemma for f1resrcmplf1d 35101. (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 6737 . . . . . 6 (𝜑𝐹 Fn 𝐴)
4 fnfvima 7253 . . . . . 6 ((𝐹 Fn 𝐴𝐶𝐴𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
53, 4syl3an1 1164 . . . . 5 ((𝜑𝐶𝐴𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
61, 5syl3an2 1165 . . . 4 ((𝜑𝜑𝑋𝐶) → (𝐹𝑋) ∈ (𝐹𝐶))
763anidm12 1421 . . 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 1164 . . . . 5 ((𝜑𝐷𝐴𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
129, 11syl3an2 1165 . . . 4 ((𝜑𝜑𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
13123anidm12 1421 . . 3 ((𝜑𝑌𝐷) → (𝐹𝑌) ∈ (𝐹𝐷))
1413ex 412 . 2 (𝜑 → (𝑌𝐷 → (𝐹𝑌) ∈ (𝐹𝐷)))
15 f1resrcmplf1dlem.4 . . . . 5 (𝜑 → ((𝐹𝐶) ∩ (𝐹𝐷)) = ∅)
16 disjne 4455 . . . . 5 ((((𝐹𝐶) ∩ (𝐹𝐷)) = ∅ ∧ (𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌))
1715, 16syl3an1 1164 . . . 4 ((𝜑 ∧ (𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌))
18173expib 1123 . . 3 (𝜑 → (((𝐹𝑋) ∈ (𝐹𝐶) ∧ (𝐹𝑌) ∈ (𝐹𝐷)) → (𝐹𝑋) ≠ (𝐹𝑌)))
19 neneq 2946 . . . 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 1540  wcel 2108  wne 2940  cin 3950  wss 3951  c0 4333  cima 5688   Fn wfn 6556  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  f1resrcmplf1d  35101
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