Theorem updjudhcoinrg 6974

Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.)

Hypotheses

Ref	Expression
updjud.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
updjud.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
updjudhf.h	⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))

Assertion

Ref	Expression
updjudhcoinrg	⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝐹   𝑥,𝐺

Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem updjudhcoinrg

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		updjud.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
2		updjud.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
3		updjudhf.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))
4	1, 2, 3	updjudhf 6972	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5		ffn 5280	. . . 4 ⊢ (𝐻:(𝐴 ⊔ 𝐵)⟶𝐶 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6	4, 5	syl 14	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
7		inrresf1 6955	. . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵)
8		f1fn 5338	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
9	7, 8	mp1i 10	. . 3 ⊢ (𝜑 → (inr ↾ 𝐵) Fn 𝐵)
10		f1f 5336	. . . . 5 ⊢ ((inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵) → (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵))
11	7, 10	ax-mp 5	. . . 4 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)
12		frn 5289	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) → ran (inr ↾ 𝐵) ⊆ (𝐴 ⊔ 𝐵))
13	11, 12	mp1i 10	. . 3 ⊢ (𝜑 → ran (inr ↾ 𝐵) ⊆ (𝐴 ⊔ 𝐵))
14		fnco 5239	. . 3 ⊢ ((𝐻 Fn (𝐴 ⊔ 𝐵) ∧ (inr ↾ 𝐵) Fn 𝐵 ∧ ran (inr ↾ 𝐵) ⊆ (𝐴 ⊔ 𝐵)) → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
15	6, 9, 13, 14	syl3anc 1217	. 2 ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
16		ffn 5280	. . 3 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺 Fn 𝐵)
17	2, 16	syl 14	. 2 ⊢ (𝜑 → 𝐺 Fn 𝐵)
18		fvco2 5498	. . . 4 ⊢ (((inr ↾ 𝐵) Fn 𝐵 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
19	9, 18	sylan 281	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
20		fvres 5453	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
21	20	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
22	21	fveq2d 5433	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐻‘(inr‘𝑏)))
23	3	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))))
24		fveq2 5429	. . . . . . . . 9 ⊢ (𝑥 = (inr‘𝑏) → (1st ‘𝑥) = (1st ‘(inr‘𝑏)))
25	24	eqeq1d 2149	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → ((1st ‘𝑥) = ∅ ↔ (1st ‘(inr‘𝑏)) = ∅))
26		fveq2 5429	. . . . . . . . 9 ⊢ (𝑥 = (inr‘𝑏) → (2nd ‘𝑥) = (2nd ‘(inr‘𝑏)))
27	26	fveq2d 5433	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐹‘(2nd ‘𝑥)) = (𝐹‘(2nd ‘(inr‘𝑏))))
28	26	fveq2d 5433	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐺‘(2nd ‘𝑥)) = (𝐺‘(2nd ‘(inr‘𝑏))))
29	25, 27, 28	ifbieq12d 3503	. . . . . . 7 ⊢ (𝑥 = (inr‘𝑏) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))))
30	29	adantl 275	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))))
31		1stinr 6969	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐵 → (1st ‘(inr‘𝑏)) = 1o)
32		1n0 6337	. . . . . . . . . . . 12 ⊢ 1o ≠ ∅
33	32	neii 2311	. . . . . . . . . . 11 ⊢ ¬ 1o = ∅
34		eqeq1 2147	. . . . . . . . . . 11 ⊢ ((1st ‘(inr‘𝑏)) = 1o → ((1st ‘(inr‘𝑏)) = ∅ ↔ 1o = ∅))
35	33, 34	mtbiri 665	. . . . . . . . . 10 ⊢ ((1st ‘(inr‘𝑏)) = 1o → ¬ (1st ‘(inr‘𝑏)) = ∅)
36	31, 35	syl 14	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐵 → ¬ (1st ‘(inr‘𝑏)) = ∅)
37	36	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ (1st ‘(inr‘𝑏)) = ∅)
38	37	adantr 274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → ¬ (1st ‘(inr‘𝑏)) = ∅)
39	38	iffalsed 3489	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))) = (𝐺‘(2nd ‘(inr‘𝑏))))
40	30, 39	eqtrd 2173	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = (𝐺‘(2nd ‘(inr‘𝑏))))
41		djurcl 6945	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → (inr‘𝑏) ∈ (𝐴 ⊔ 𝐵))
42	41	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (inr‘𝑏) ∈ (𝐴 ⊔ 𝐵))
43	2	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐺:𝐵⟶𝐶)
44		2ndinr 6970	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 → (2nd ‘(inr‘𝑏)) = 𝑏)
45	44	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2nd ‘(inr‘𝑏)) = 𝑏)
46		simpr 109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
47	45, 46	eqeltrd 2217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2nd ‘(inr‘𝑏)) ∈ 𝐵)
48	43, 47	ffvelrnd 5564	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2nd ‘(inr‘𝑏))) ∈ 𝐶)
49	23, 40, 42, 48	fvmptd 5510	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘(inr‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
50	22, 49	eqtrd 2173	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
51	45	fveq2d 5433	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2nd ‘(inr‘𝑏))) = (𝐺‘𝑏))
52	19, 50, 51	3eqtrd 2177	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐺‘𝑏))
53	15, 17, 52	eqfnfvd 5529	1 ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481   ⊆ wss 3076  ∅c0 3368  ifcif 3479   ↦ cmpt 3997  ran crn 4548   ↾ cres 4549   ∘ ccom 4551   Fn wfn 5126  ⟶wf 5127  –1-1→wf1 5128  ‘cfv 5131  1^st c1st 6044  2^nd c2nd 6045  1_oc1o 6314   ⊔ cdju 6930  inrcinr 6939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-dju 6931  df-inl 6940  df-inr 6941

This theorem is referenced by:  updjud  6975