Theorem updjudhcoinrg 9869

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 9867	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5	4	ffnd 6669	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6		inrresf 9852	. . . 4 ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)
7		ffn 6668	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
8	6, 7	mp1i 13	. . 3 ⊢ (𝜑 → (inr ↾ 𝐵) Fn 𝐵)
9		frn 6675	. . . 4 ⊢ ((inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) → ran (inr ↾ 𝐵) ⊆ (𝐴 ⊔ 𝐵))
10	6, 9	mp1i 13	. . 3 ⊢ (𝜑 → ran (inr ↾ 𝐵) ⊆ (𝐴 ⊔ 𝐵))
11		fnco 6618	. . 3 ⊢ ((𝐻 Fn (𝐴 ⊔ 𝐵) ∧ (inr ↾ 𝐵) Fn 𝐵 ∧ ran (inr ↾ 𝐵) ⊆ (𝐴 ⊔ 𝐵)) → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
12	5, 8, 10, 11	syl3anc 1371	. 2 ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) Fn 𝐵)
13	2	ffnd 6669	. 2 ⊢ (𝜑 → 𝐺 Fn 𝐵)
14		fvco2 6938	. . . 4 ⊢ (((inr ↾ 𝐵) Fn 𝐵 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
15	8, 14	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐻‘((inr ↾ 𝐵)‘𝑏)))
16		fvres 6861	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
17	16	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((inr ↾ 𝐵)‘𝑏) = (inr‘𝑏))
18	17	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐻‘(inr‘𝑏)))
19		fveqeq2 6851	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → ((1st ‘𝑥) = ∅ ↔ (1st ‘(inr‘𝑏)) = ∅))
20		2fveq3 6847	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐹‘(2nd ‘𝑥)) = (𝐹‘(2nd ‘(inr‘𝑏))))
21		2fveq3 6847	. . . . . . . 8 ⊢ (𝑥 = (inr‘𝑏) → (𝐺‘(2nd ‘𝑥)) = (𝐺‘(2nd ‘(inr‘𝑏))))
22	19, 20, 21	ifbieq12d 4514	. . . . . . 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)))))
23	22	adantl 482	. . . . . 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)))))
24		1stinr 9865	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐵 → (1st ‘(inr‘𝑏)) = 1o)
25		1n0 8434	. . . . . . . . . . . 12 ⊢ 1o ≠ ∅
26	25	neii 2945	. . . . . . . . . . 11 ⊢ ¬ 1o = ∅
27		eqeq1 2740	. . . . . . . . . . 11 ⊢ ((1st ‘(inr‘𝑏)) = 1o → ((1st ‘(inr‘𝑏)) = ∅ ↔ 1o = ∅))
28	26, 27	mtbiri 326	. . . . . . . . . 10 ⊢ ((1st ‘(inr‘𝑏)) = 1o → ¬ (1st ‘(inr‘𝑏)) = ∅)
29	24, 28	syl 17	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝐵 → ¬ (1st ‘(inr‘𝑏)) = ∅)
30	29	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ (1st ‘(inr‘𝑏)) = ∅)
31	30	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → ¬ (1st ‘(inr‘𝑏)) = ∅)
32	31	iffalsed 4497	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘(inr‘𝑏)) = ∅, (𝐹‘(2nd ‘(inr‘𝑏))), (𝐺‘(2nd ‘(inr‘𝑏)))) = (𝐺‘(2nd ‘(inr‘𝑏))))
33	23, 32	eqtrd 2776	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 = (inr‘𝑏)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = (𝐺‘(2nd ‘(inr‘𝑏))))
34		djurcl 9847	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 → (inr‘𝑏) ∈ (𝐴 ⊔ 𝐵))
35	34	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (inr‘𝑏) ∈ (𝐴 ⊔ 𝐵))
36	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐺:𝐵⟶𝐶)
37		2ndinr 9866	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 → (2nd ‘(inr‘𝑏)) = 𝑏)
38	37	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2nd ‘(inr‘𝑏)) = 𝑏)
39		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
40	38, 39	eqeltrd 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (2nd ‘(inr‘𝑏)) ∈ 𝐵)
41	36, 40	ffvelcdmd 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2nd ‘(inr‘𝑏))) ∈ 𝐶)
42	3, 33, 35, 41	fvmptd2 6956	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘(inr‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
43	18, 42	eqtrd 2776	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐻‘((inr ↾ 𝐵)‘𝑏)) = (𝐺‘(2nd ‘(inr‘𝑏))))
44	38	fveq2d 6846	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐺‘(2nd ‘(inr‘𝑏))) = (𝐺‘𝑏))
45	15, 43, 44	3eqtrd 2780	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝐻 ∘ (inr ↾ 𝐵))‘𝑏) = (𝐺‘𝑏))
46	12, 13, 45	eqfnfvd 6985	1 ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ⊆ wss 3910  ∅c0 4282  ifcif 4486   ↦ cmpt 5188  ran crn 5634   ↾ cres 5635   ∘ ccom 5637   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  1^st c1st 7919  2^nd c2nd 7920  1_oc1o 8405   ⊔ cdju 9834  inrcinr 9836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-dju 9837  df-inr 9839

This theorem is referenced by:  updjud  9870