Theorem focofo 6760

Description: Composition of onto functions. Generalisation of foco 6761. (Contributed by AV, 29-Sep-2024.)

Assertion

Ref	Expression
focofo	⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵)

Proof of Theorem focofo

Step	Hyp	Ref	Expression
1		fof 6747	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		fcof 6686	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
3	1, 2	sylan 581	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
4	3	3adant3 1133	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
5		rnco 6211	. . 3 ⊢ ran (𝐹 ∘ 𝐺) = ran (𝐹 ↾ ran 𝐺)
6	1	freld 6669	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Rel 𝐹)
7	6	3ad2ant1 1134	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → Rel 𝐹)
8		fdm 6672	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
9	8	eqcomd 2743	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
10	1, 9	syl 17	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐴 = dom 𝐹)
11	10	sseq1d 3966	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐴 ⊆ ran 𝐺 ↔ dom 𝐹 ⊆ ran 𝐺))
12	11	biimpa 476	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ⊆ ran 𝐺) → dom 𝐹 ⊆ ran 𝐺)
13		relssres 5982	. . . . . 6 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → (𝐹 ↾ ran 𝐺) = 𝐹)
14	13	rneqd 5888	. . . . 5 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
15	7, 12, 14	3imp3i2an 1347	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
16		forn 6750	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
17	16	3ad2ant1 1134	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran 𝐹 = 𝐵)
18	15, 17	eqtrd 2772	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = 𝐵)
19	5, 18	eqtrid 2784	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ∘ 𝐺) = 𝐵)
20		dffo2 6751	. 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵 ↔ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵 ∧ ran (𝐹 ∘ 𝐺) = 𝐵))
21	4, 19, 20	sylanbrc 584	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ⊆ wss 3902  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Rel wrel 5630  Fun wfun 6487  ⟶wf 6489  –onto→wfo 6491

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

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-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499

This theorem is referenced by:  foco  6761  funfocofob  47391