Theorem focofo 6806

Description: Composition of onto functions. Generalisation of foco 6807. (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 6793	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		fcof 6730	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
3	1, 2	sylan 592	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
4	3	3adant3 1150	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
5		rnco 6252	. . 3 ⊢ ran (𝐹 ∘ 𝐺) = ran (𝐹 ↾ ran 𝐺)
6	1	freld 6713	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Rel 𝐹)
7	6	3ad2ant1 1151	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → Rel 𝐹)
8		fdm 6716	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
9	8	eqcomd 2768	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
10	1, 9	syl 18	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐴 = dom 𝐹)
11	10	sseq1d 3965	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐴 ⊆ ran 𝐺 ↔ dom 𝐹 ⊆ ran 𝐺))
12	11	biimpa 482	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ⊆ ran 𝐺) → dom 𝐹 ⊆ ran 𝐺)
13		relssres 6019	. . . . . 6 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → (𝐹 ↾ ran 𝐺) = 𝐹)
14	13	rneqd 5926	. . . . 5 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
15	7, 12, 14	3imp3i2an 1364	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
16		forn 6796	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
17	16	3ad2ant1 1151	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran 𝐹 = 𝐵)
18	15, 17	eqtrd 2797	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = 𝐵)
19	5, 18	eqtrid 2809	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ∘ 𝐺) = 𝐵)
20		dffo2 6797	. 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵 ↔ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵 ∧ ran (𝐹 ∘ 𝐺) = 𝐵))
21	4, 19, 20	sylanbrc 595	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ⊆ wss 3902  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662   ∘ ccom 5663  Rel wrel 5664  Fun wfun 6531  ⟶wf 6533  –onto→wfo 6535

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-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-nfc 2911  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-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-fun 6539  df-fn 6540  df-f 6541  df-fo 6543

This theorem is used by:  foco  6807  funfocofob  47974