Theorem imaf1homlem 50018

Description: Lemma for imaf1hom 50019 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 503	. . . 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 6071	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
8		imaf1hom.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
9		imaf1hom.s	. . . . 5 ⊢ 𝑆 = (𝐹 “ 𝐴)
10	8, 9	eleqtrdi 2872	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐹 “ 𝐴))
11	7, 10	sselid 3932	. . 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 596	. 2 ⊢ (𝜑 → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
14		f1ocnvfv2 7281	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
15	3, 11, 14	syl2anc 596	. 2 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
16		f1ocnvdm 7289	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→ran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → (◡𝐹‘𝑋) ∈ 𝐵)
17	3, 11, 16	syl2anc 596	. 2 ⊢ (𝜑 → (◡𝐹‘𝑋) ∈ 𝐵)
18	13, 15, 17	3jca 1146	1 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {csn 4587  ^◡ccnv 5658  ran crn 5660   “ cima 5662   Fn wfn 6532  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 used by:  imaf1hom  50019  imaf1co  50066