Theorem foco 5518

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 5511	. . 3 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶))
2		dffo2 5511	. . 3 ⊢ (𝐺:𝐴–onto→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵))
3		fco 5448	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
4	3	ad2ant2r 509	. . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
5		fdm 5438	. . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐶 → dom 𝐹 = 𝐵)
6		eqtr3 2226	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
7	5, 6	sylan 283	. . . . . . 7 ⊢ ((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
8		rncoeq 4958	. . . . . . . . 9 ⊢ (dom 𝐹 = ran 𝐺 → ran (𝐹 ∘ 𝐺) = ran 𝐹)
9	8	eqeq1d 2215	. . . . . . . 8 ⊢ (dom 𝐹 = ran 𝐺 → (ran (𝐹 ∘ 𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
10	9	biimpar 297	. . . . . . 7 ⊢ ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
11	7, 10	sylan 283	. . . . . 6 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹 ∘ 𝐺) = 𝐶)
12	11	an32s 568	. . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹 ∘ 𝐺) = 𝐶)
13	12	adantrl 478	. . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴⟶𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹 ∘ 𝐺) = 𝐶)
14	4, 13	jca 306	. . 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 291	. 2 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
16		dffo2 5511	. 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ ran (𝐹 ∘ 𝐺) = 𝐶))
17	15, 16	sylibr 134	1 ⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  dom cdm 4680  ran crn 4681   ∘ ccom 4684  ⟶wf 5273  –onto→wfo 5275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-br 4049  df-opab 4111  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-fun 5279  df-fn 5280  df-f 5281  df-fo 5283

This theorem is referenced by:  f1oco  5554  nninfct  12412  ennnfonelemnn0  12843  ctinfomlemom  12848  qnnen  12852  enctlem  12853