Theorem focofo 6809

Description: Composition of onto functions. Generalisation of foco 6810. (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 6796	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		fcof 6733	. . . 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 6253	. . 3 ⊢ ran (𝐹 ∘ 𝐺) = ran (𝐹 ↾ ran 𝐺)
6	1	freld 6716	. . . . . 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 6719	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
9	8	eqcomd 2766	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
10	1, 9	syl 18	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐴 = dom 𝐹)
11	10	sseq1d 3939	. . . . . 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 6010	. . . . . 6 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → (𝐹 ↾ ran 𝐺) = 𝐹)
14	13	rneqd 5919	. . . . 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 6799	. . . . 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 2795	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = 𝐵)
19	5, 18	eqtrid 2807	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ∘ 𝐺) = 𝐵)
20		dffo2 6800	. 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 1569   ⊆ wss 3895  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   ↾ cres 5652   “ cima 5653   ∘ ccom 5654  Rel wrel 5655  Fun wfun 6532  ⟶wf 6534  –onto→wfo 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544

This theorem is used by:  foco  6810  funfocofob  48148