Theorem imaf1homlem 49804

Description: Lemma for imaf1hom 49805 and other theorems. (Contributed by Zhi Wang, 7-Nov-2025.)

Hypotheses

Ref	Expression
imaf1hom.s	⊢ 𝑆 = (𝐹 “ 𝐴)
imaf1hom.1	⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)
imaf1hom.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)

Assertion

Ref	Expression
imaf1homlem	⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))

Proof of Theorem imaf1homlem

Step	Hyp	Ref	Expression
1		imaf1hom.1	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)
2		f1f1orn 6833	. . . . 5 ⊢ (𝐹:𝐵–1-1→𝐶 → 𝐹:𝐵–1-1-onto→ran 𝐹)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→ran 𝐹)
4		dff1o4 6830	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn ran 𝐹))
5	4	simprbi 502	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→ran 𝐹 → ◡𝐹 Fn ran 𝐹)
6	3, 5	syl 18	. . 3 ⊢ (𝜑 → ◡𝐹 Fn ran 𝐹)
7		imassrn 6074	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
8		imaf1hom.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
9		imaf1hom.s	. . . . 5 ⊢ 𝑆 = (𝐹 “ 𝐴)
10	8, 9	eleqtrdi 2879	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐹 “ 𝐴))
11	7, 10	sselid 3943	. . 3 ⊢ (𝜑 → 𝑋 ∈ ran 𝐹)
12		fnsnfv 6961	. . 3 ⊢ ((◡𝐹 Fn ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
13	6, 11, 12	syl2anc 595	. 2 ⊢ (𝜑 → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
14		f1ocnvfv2 7276	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
15	3, 11, 14	syl2anc 595	. 2 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
16		f1ocnvdm 7284	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → (◡𝐹‘𝑋) ∈ 𝐵)
17	3, 11, 16	syl2anc 595	. 2 ⊢ (𝜑 → (◡𝐹‘𝑋) ∈ 𝐵)
18	13, 15, 17	3jca 1144	1 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {csn 4594  ^◡ccnv 5661  ran crn 5663   “ cima 5665   Fn wfn 6532  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by:  imaf1hom  49805  imaf1co  49852