Theorem fimadmfoALT 6757

Description: Alternate proof of fimadmfo 6755, based on fores 6756. A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
fimadmfoALT	⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴))

Proof of Theorem fimadmfoALT

Step	Hyp	Ref	Expression
1		fdm 6671	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2		frel 6667	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
3		resdm 5985	. . . . . 6 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
4	3	eqcomd 2743	. . . . 5 ⊢ (Rel 𝐹 → 𝐹 = (𝐹 ↾ dom 𝐹))
5	2, 4	syl 17	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ dom 𝐹))
6		reseq2 5933	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ↾ dom 𝐹) = (𝐹 ↾ 𝐴))
7	5, 6	sylan9eq 2792	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ dom 𝐹 = 𝐴) → 𝐹 = (𝐹 ↾ 𝐴))
8	1, 7	mpdan 688	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝐹 ↾ 𝐴))
9		ffun 6665	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
10		eqimss2 3982	. . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹)
11	1, 10	syl 17	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
12	9, 11	jca 511	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
13	12	adantr 480	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
14		fores 6756	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))
15	13, 14	syl 17	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))
16		foeq1 6742	. . . 4 ⊢ (𝐹 = (𝐹 ↾ 𝐴) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)))
17	16	adantl 481	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → (𝐹:𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)))
18	15, 17	mpbird 257	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐹 = (𝐹 ↾ 𝐴)) → 𝐹:𝐴–onto→(𝐹 “ 𝐴))
19	8, 18	mpdan 688	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ⊆ wss 3890  dom cdm 5624   ↾ cres 5626   “ cima 5627  Rel wrel 5629  Fun wfun 6486  ⟶wf 6488  –onto→wfo 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  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-fun 6494  df-fn 6495  df-f 6496  df-fo 6498

This theorem is referenced by: (None)