Theorem f1resrcmplf1dlem 35267

Description: Lemma for f1resrcmplf1d 35268. (Contributed by BTernaryTau, 27-Sep-2023.)

Hypotheses

Ref	Expression
f1resrcmplf1dlem.1	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
f1resrcmplf1dlem.2	⊢ (𝜑 → 𝐷 ⊆ 𝐴)
f1resrcmplf1dlem.3	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
f1resrcmplf1dlem.4	⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅)

Assertion

Ref	Expression
f1resrcmplf1dlem	⊢ (𝜑 → ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))

Proof of Theorem f1resrcmplf1dlem

Step	Hyp	Ref	Expression
1		f1resrcmplf1dlem.1	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
2		f1resrcmplf1dlem.3	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	ffnd 6656	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		fnfvima 7177	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))
5	3, 4	syl3an1 1169	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))
6	1, 5	syl3an2 1170	. . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))
7	6	3anidm12 1427	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))
8	7	ex 413	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝐶 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)))
9		f1resrcmplf1dlem.2	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
10		fnfvima 7177	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))
11	3, 10	syl3an1 1169	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))
12	9, 11	syl3an2 1170	. . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))
13	12	3anidm12 1427	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))
14	13	ex 413	. 2 ⊢ (𝜑 → (𝑌 ∈ 𝐷 → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)))
15		f1resrcmplf1dlem.4	. . . . 5 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅)
16		disjne 4383	. . . . 5 ⊢ ((((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅ ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))
17	15, 16	syl3an1 1169	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))
18	17	3expib 1128	. . 3 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)))
19		neneq 2940	. . . 4 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) → ¬ (𝐹‘𝑋) = (𝐹‘𝑌))
20	19	pm2.21d 121	. . 3 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
21	18, 20	syl6 35	. 2 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
22	8, 14, 21	syl2and 614	1 ⊢ (𝜑 → ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261   “ cima 5621   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493

This theorem is referenced by:  f1resrcmplf1d  35268