Theorem f1resrcmplf1dlem 7309

Description: Lemma for f1resrcmplf1d 7310. (Contributed by BTernaryTau, 27-Sep-2023.) (Revised by Mingli Yuan, 15-Aug-2026.)

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)) = ∅)
f1resrcmplf1dlem.x	⊢ (𝜑 → 𝑋 ∈ 𝐶)
f1resrcmplf1dlem.y	⊢ (𝜑 → 𝑌 ∈ 𝐷)
f1resrcmplf1dlem.5	⊢ (𝜑 → (𝐹‘𝑋) = (𝐹‘𝑌))

Assertion

Ref	Expression
f1resrcmplf1dlem	⊢ (𝜑 → 𝑋 = 𝑌)

Proof of Theorem f1resrcmplf1dlem

Step	Hyp	Ref	Expression
1		f1resrcmplf1dlem.5	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐹‘𝑌))
2		f1resrcmplf1dlem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐶)
3		f1resrcmplf1dlem.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
4	2, 3	jca 521	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷))
5		f1resrcmplf1dlem.1	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
6		f1resrcmplf1dlem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7	6	ffnd 6740	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝐴)
8		fnfvima 7270	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))
9	7, 8	syl3an1 1181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))
10	5, 9	syl3an2 1182	. . . . . 6 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))
11	10	3anidm12 1447	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶))
12	11	ex 418	. . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐶 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐶)))
13		f1resrcmplf1dlem.2	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
14		fnfvima 7270	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))
15	7, 14	syl3an1 1181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐴 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))
16	13, 15	syl3an2 1182	. . . . . 6 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))
17	16	3anidm12 1447	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷))
18	17	ex 418	. . . 4 ⊢ (𝜑 → (𝑌 ∈ 𝐷 → (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)))
19		f1resrcmplf1dlem.4	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅)
20		disjne 4411	. . . . . . 7 ⊢ ((((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅ ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))
21	19, 20	syl3an1 1181	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))
22	21	3expib 1140	. . . . 5 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌)))
23		neneq 2965	. . . . . 6 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) → ¬ (𝐹‘𝑋) = (𝐹‘𝑌))
24	23	pm2.21d 122	. . . . 5 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
25	22, 24	syl6 36	. . . 4 ⊢ (𝜑 → (((𝐹‘𝑋) ∈ (𝐹 “ 𝐶) ∧ (𝐹‘𝑌) ∈ (𝐹 “ 𝐷)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
26	12, 18, 25	syl2and 620	. . 3 ⊢ (𝜑 → ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
27	4, 26	mpd 16	. 2 ⊢ (𝜑 → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌))
28	1, 27	mpd 16	1 ⊢ (𝜑 → 𝑋 = 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281   “ cima 5661   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574

This theorem is used by:  f1resrcmplf1d  7310