Theorem f1cof1blem 47086

Description: Lemma for f1cof1b 47089 and focofob 47092. (Contributed by AV, 18-Sep-2024.)

Hypotheses

Ref	Expression
fcores.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fcores.e	⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
fcores.p	⊢ 𝑃 = (◡𝐹 “ 𝐶)
fcores.x	⊢ 𝑋 = (𝐹 ↾ 𝑃)
fcores.g	⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
fcores.y	⊢ 𝑌 = (𝐺 ↾ 𝐸)
f1cof1blem.s	⊢ (𝜑 → ran 𝐹 = 𝐶)

Assertion

Ref	Expression
f1cof1blem	⊢ (𝜑 → ((𝑃 = 𝐴 ∧ 𝐸 = 𝐶) ∧ (𝑋 = 𝐹 ∧ 𝑌 = 𝐺)))

Proof of Theorem f1cof1blem

Step	Hyp	Ref	Expression
1		fcores.p	. . . . 5 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
2		f1cof1blem.s	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐶)
3	2	eqcomd 2743	. . . . . 6 ⊢ (𝜑 → 𝐶 = ran 𝐹)
4	3	imaeq2d 6078	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝐶) = (◡𝐹 “ ran 𝐹))
5	1, 4	eqtrid 2789	. . . 4 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ ran 𝐹))
6		cnvimarndm 6101	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
7		fcores.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8	7	fdmd 6746	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
9	6, 8	eqtrid 2789	. . . 4 ⊢ (𝜑 → (◡𝐹 “ ran 𝐹) = 𝐴)
10	5, 9	eqtrd 2777	. . 3 ⊢ (𝜑 → 𝑃 = 𝐴)
11		fcores.e	. . . 4 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
12		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ran 𝐹 = 𝐶) → ran 𝐹 = 𝐶)
13	12	ineq1d 4219	. . . . . 6 ⊢ ((𝜑 ∧ ran 𝐹 = 𝐶) → (ran 𝐹 ∩ 𝐶) = (𝐶 ∩ 𝐶))
14		inidm 4227	. . . . . 6 ⊢ (𝐶 ∩ 𝐶) = 𝐶
15	13, 14	eqtrdi 2793	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 = 𝐶) → (ran 𝐹 ∩ 𝐶) = 𝐶)
16	2, 15	mpdan 687	. . . 4 ⊢ (𝜑 → (ran 𝐹 ∩ 𝐶) = 𝐶)
17	11, 16	eqtrid 2789	. . 3 ⊢ (𝜑 → 𝐸 = 𝐶)
18	10, 17	jca 511	. 2 ⊢ (𝜑 → (𝑃 = 𝐴 ∧ 𝐸 = 𝐶))
19		fcores.x	. . . 4 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
20	5, 6	eqtrdi 2793	. . . . 5 ⊢ (𝜑 → 𝑃 = dom 𝐹)
21	20	reseq2d 5997	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑃) = (𝐹 ↾ dom 𝐹))
22	19, 21	eqtrid 2789	. . 3 ⊢ (𝜑 → 𝑋 = (𝐹 ↾ dom 𝐹))
23	7	freld 6742	. . . 4 ⊢ (𝜑 → Rel 𝐹)
24		resdm 6044	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
25	23, 24	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ↾ dom 𝐹) = 𝐹)
26	22, 25	eqtrd 2777	. 2 ⊢ (𝜑 → 𝑋 = 𝐹)
27		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
28		fcores.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
29	28	fdmd 6746	. . . . . 6 ⊢ (𝜑 → dom 𝐺 = 𝐶)
30	17, 29	eqtr4d 2780	. . . . 5 ⊢ (𝜑 → 𝐸 = dom 𝐺)
31	30	reseq2d 5997	. . . 4 ⊢ (𝜑 → (𝐺 ↾ 𝐸) = (𝐺 ↾ dom 𝐺))
32	27, 31	eqtrid 2789	. . 3 ⊢ (𝜑 → 𝑌 = (𝐺 ↾ dom 𝐺))
33	28	freld 6742	. . . 4 ⊢ (𝜑 → Rel 𝐺)
34		resdm 6044	. . . 4 ⊢ (Rel 𝐺 → (𝐺 ↾ dom 𝐺) = 𝐺)
35	33, 34	syl 17	. . 3 ⊢ (𝜑 → (𝐺 ↾ dom 𝐺) = 𝐺)
36	32, 35	eqtrd 2777	. 2 ⊢ (𝜑 → 𝑌 = 𝐺)
37	18, 26, 36	jca32 515	1 ⊢ (𝜑 → ((𝑃 = 𝐴 ∧ 𝐸 = 𝐶) ∧ (𝑋 = 𝐹 ∧ 𝑌 = 𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∩ cin 3950  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688  Rel wrel 5690  ⟶wf 6557

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-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-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  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-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565

This theorem is referenced by:  f1cof1b  47089