Theorem focofo 6828

Description: Composition of onto functions. Generalisation of foco 6829. (Contributed by AV, 29-Sep-2024.)

Assertion

Ref	Expression
focofo	⊢ ((𝐹:𝐴⟶onto𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶onto𝐵)

Proof of Theorem focofo

Step	Hyp	Ref	Expression
1		fof 6815	. . . 4 ⊢ (𝐹:𝐴⟶onto𝐵 → 𝐹:𝐴⟶𝐵)
2		fcof 6752	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
3	1, 2	sylan 592	. . 3 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
4	3	3adant3 1150	. 2 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
5		rnco 6265	. . 3 ⊢ ran (𝐹 ∘ 𝐺) = ran (𝐹 ↾ ran 𝐺)
6	1	freld 6735	. . . . . 6 ⊢ (𝐹:𝐴⟶onto𝐵 → Rel 𝐹)
7	6	3ad2ant1 1151	. . . . 5 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → Rel 𝐹)
8		fdm 6738	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
9	8	eqcomd 2769	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
10	1, 9	syl 18	. . . . . . 7 ⊢ (𝐹:𝐴⟶onto𝐵 → 𝐴 = dom 𝐹)
11	10	sseq1d 3944	. . . . . 6 ⊢ (𝐹:𝐴⟶onto𝐵 → (𝐴 ⊆ ran 𝐺 ↔ dom 𝐹 ⊆ ran 𝐺))
12	11	biimpa 482	. . . . 5 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐴 ⊆ ran 𝐺) → dom 𝐹 ⊆ ran 𝐺)
13		relssres 6022	. . . . . 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⟶onto𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
16		forn 6818	. . . . 5 ⊢ (𝐹:𝐴⟶onto𝐵 → ran 𝐹 = 𝐵)
17	16	3ad2ant1 1151	. . . 4 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran 𝐹 = 𝐵)
18	15, 17	eqtrd 2798	. . 3 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = 𝐵)
19	5, 18	eqtrid 2810	. 2 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ∘ 𝐺) = 𝐵)
20		dffo2 6819	. 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶onto𝐵 ↔ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵 ∧ ran (𝐹 ∘ 𝐺) = 𝐵))
21	4, 19, 20	sylanbrc 595	1 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶onto𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ⊆ wss 3900  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658   ∘ ccom 5659  Rel wrel 5660  Fun wfun 6549  ⟶wf 6551  ⟶ontowfo 6553

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-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-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  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-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-fun 6557  df-fn 6558  df-f 6559  df-fo 6561

This theorem is used by:  foco  6829  funfocofob  48351