Theorem imaf1homlem 49352

Description: Lemma for imaf1hom 49353 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 6785	. . . . 5 ⊢ (𝐹:𝐵–1-1→𝐶 → 𝐹:𝐵–1-1-onto→ran 𝐹)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→ran 𝐹)
4		dff1o4 6782	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn ran 𝐹))
5	4	simprbi 496	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→ran 𝐹 → ◡𝐹 Fn ran 𝐹)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → ◡𝐹 Fn ran 𝐹)
7		imassrn 6030	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
8		imaf1hom.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
9		imaf1hom.s	. . . . 5 ⊢ 𝑆 = (𝐹 “ 𝐴)
10	8, 9	eleqtrdi 2846	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐹 “ 𝐴))
11	7, 10	sselid 3931	. . 3 ⊢ (𝜑 → 𝑋 ∈ ran 𝐹)
12		fnsnfv 6913	. . 3 ⊢ ((◡𝐹 Fn ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
13	6, 11, 12	syl2anc 584	. 2 ⊢ (𝜑 → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
14		f1ocnvfv2 7223	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
15	3, 11, 14	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
16		f1ocnvdm 7231	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → (◡𝐹‘𝑋) ∈ 𝐵)
17	3, 11, 16	syl2anc 584	. 2 ⊢ (𝜑 → (◡𝐹‘𝑋) ∈ 𝐵)
18	13, 15, 17	3jca 1128	1 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {csn 4580  ^◡ccnv 5623  ran crn 5625   “ cima 5627   Fn wfn 6487  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  imaf1hom  49353  imaf1co  49400