Theorem imaf1homlem 50426

Description: Lemma for imaf1hom 50427 and other theorems. (Contributed by Zhi Wang, 7-Nov-2025.)

Hypotheses

Ref	Expression
imaf1hom.s	⊢ 𝑆 = (𝐹 “ 𝐴)
imaf1hom.1	⊢ (𝜑 → 𝐹:𝐵⟶1-1𝐶)
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⟶1-1𝐶)
2		f1f1orn 6855	. . . . 5 ⊢ (𝐹:𝐵⟶1-1𝐶 → 𝐹:𝐵⟶1-1ontoran 𝐹)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶1-1ontoran 𝐹)
4		dff1o4 6852	. . . . 5 ⊢ (𝐹:𝐵⟶1-1ontoran 𝐹 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn ran 𝐹))
5	4	simprbi 503	. . . 4 ⊢ (𝐹:𝐵⟶1-1ontoran 𝐹 → ◡𝐹 Fn ran 𝐹)
6	3, 5	syl 18	. . 3 ⊢ (𝜑 → ◡𝐹 Fn ran 𝐹)
7		imassrn 6209	. . . 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 3911	. . 3 ⊢ (𝜑 → 𝑋 ∈ ran 𝐹)
12		fnsnfv 6984	. . 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 7305	. . 3 ⊢ ((𝐹:𝐵⟶1-1ontoran 𝐹 ∧ 𝑋 ∈ ran 𝐹) → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
15	3, 11, 14	syl2anc 596	. 2 ⊢ (𝜑 → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
16		f1ocnvdm 7313	. . 3 ⊢ ((𝐹:𝐵⟶1-1ontoran 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 2147  {csn 4588  ^◡ccnv 5654  ran crn 5656   “ cima 5658   Fn wfn 6550  ⟶1-1wf1 6552  ⟶1-1ontowf1o 6554  ‘cfv 6555

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563

This theorem is used by:  imaf1hom  50427  imaf1co  50474