Theorem focofo 6785

Description: Composition of onto functions. Generalisation of foco 6786. (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 6772	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		fcof 6711	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
3	1, 2	sylan 580	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
4	3	3adant3 1132	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
5		rnco 6225	. . 3 ⊢ ran (𝐹 ∘ 𝐺) = ran (𝐹 ↾ ran 𝐺)
6	1	freld 6694	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Rel 𝐹)
7	6	3ad2ant1 1133	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → Rel 𝐹)
8		fdm 6697	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
9	8	eqcomd 2735	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
10	1, 9	syl 17	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐴 = dom 𝐹)
11	10	sseq1d 3978	. . . . . 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 5993	. . . . . 6 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → (𝐹 ↾ ran 𝐺) = 𝐹)
14	13	rneqd 5902	. . . . 5 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
15	7, 12, 14	3imp3i2an 1346	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹)
16		forn 6775	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
17	16	3ad2ant1 1133	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran 𝐹 = 𝐵)
18	15, 17	eqtrd 2764	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = 𝐵)
19	5, 18	eqtrid 2776	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ∘ 𝐺) = 𝐵)
20		dffo2 6776	. 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵 ↔ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵 ∧ ran (𝐹 ∘ 𝐺) = 𝐵))
21	4, 19, 20	sylanbrc 583	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ⊆ wss 3914  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641   ∘ ccom 5642  Rel wrel 5643  Fun wfun 6505  ⟶wf 6507  –onto→wfo 6509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517

This theorem is referenced by:  foco  6786  funfocofob  47079