Theorem imaf1homlem 49733

Description: Lemma for imaf1hom 49734 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 6820	. . . . 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 6817	. . . . 5 ⊢ (𝐹:𝐵–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn ran 𝐹))
5	4	simprbi 501	. . . 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 6062	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
8		imaf1hom.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
9		imaf1hom.s	. . . . 5 ⊢ 𝑆 = (𝐹 “ 𝐴)
10	8, 9	eleqtrdi 2874	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐹 “ 𝐴))
11	7, 10	sselid 3936	. . 3 ⊢ (𝜑 → 𝑋 ∈ ran 𝐹)
12		fnsnfv 6948	. . 3 ⊢ ((◡𝐹 Fn ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
13	6, 11, 12	syl2anc 593	. 2 ⊢ (𝜑 → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
14		f1ocnvfv2 7263	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
15	3, 11, 14	syl2anc 593	. 2 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
16		f1ocnvdm 7271	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → (◡𝐹‘𝑋) ∈ 𝐵)
17	3, 11, 16	syl2anc 593	. 2 ⊢ (𝜑 → (◡𝐹‘𝑋) ∈ 𝐵)
18	13, 15, 17	3jca 1142	1 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {csn 4584  ^◡ccnv 5648  ran crn 5650   “ cima 5652   Fn wfn 6518  –1-1→wf1 6520  –1-1-onto→wf1o 6522  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531

This theorem is referenced by:  imaf1hom  49734  imaf1co  49781