Theorem foco 6595

Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.)

Assertion

Ref	Expression
foco	⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)

Proof of Theorem foco

Step	Hyp	Ref	Expression
1		dffo2 6587	. . 3 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶))
2		dffo2 6587	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵))
3		fco 6524	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
4	3	ad2ant2r 745	. . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
5		fdm 6515	. . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐶 → dom 𝐹 = 𝐵)
6		eqtr3 2841	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
7	5, 6	sylan 582	. . . . . . 7 ⊢ ((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
8		rncoeq 5839	. . . . . . . . 9 ⊢ (dom 𝐹 = ran 𝐺 → ran (𝐹 ∘ 𝐺) = ran 𝐹)
9	8	eqeq1d 2821	. . . . . . . 8 ⊢ (dom 𝐹 = ran 𝐺 → (ran (𝐹 ∘ 𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
10	9	biimpar 480	. . . . . . 7 ⊢ ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
11	7, 10	sylan 582	. . . . . 6 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
12	11	an32s 650	. . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹 ∘ 𝐺) = 𝐶)
13	12	adantrl 714	. . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹 ∘ 𝐺) = 𝐶)
14	4, 13	jca 514	. . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
15	1, 2, 14	syl2anb 599	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
16		dffo2 6587	. 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
17	15, 16	sylibr 236	1 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531  dom cdm 5548  ran crn 5549   ∘ ccom 5552  ⟶wf 6344  –onto→wfo 6346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354

This theorem is referenced by:  f1oco  6630  wdomtr  9031  fin1a2lem7  9820  cofull  17196  sursubmefmnd  18053  uniiccdif  24171