Theorem updjudhcoinlf 7078

Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first 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
updjudhcoinlf	⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)

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

Allowed substitution hints:   𝐺(𝑥)   𝐻(𝑥)

Proof of Theorem updjudhcoinlf

Dummy variable 𝑎 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 7077	. . . 4 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
5		ffn 5365	. . . 4 ⊢ (𝐻:(𝐴 ⊔ 𝐵)⟶𝐶 → 𝐻 Fn (𝐴 ⊔ 𝐵))
6	4, 5	syl 14	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐴 ⊔ 𝐵))
7		inlresf1 7059	. . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)
8		f1fn 5423	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) → (inl ↾ 𝐴) Fn 𝐴)
9	7, 8	mp1i 10	. . 3 ⊢ (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
10		f1f 5421	. . . . 5 ⊢ ((inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) → (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵))
11	7, 10	ax-mp 5	. . . 4 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
12		frn 5374	. . . 4 ⊢ ((inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) → ran (inl ↾ 𝐴) ⊆ (𝐴 ⊔ 𝐵))
13	11, 12	mp1i 10	. . 3 ⊢ (𝜑 → ran (inl ↾ 𝐴) ⊆ (𝐴 ⊔ 𝐵))
14		fnco 5324	. . 3 ⊢ ((𝐻 Fn (𝐴 ⊔ 𝐵) ∧ (inl ↾ 𝐴) Fn 𝐴 ∧ ran (inl ↾ 𝐴) ⊆ (𝐴 ⊔ 𝐵)) → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
15	6, 9, 13, 14	syl3anc 1238	. 2 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
16		ffn 5365	. . 3 ⊢ (𝐹:𝐴⟶𝐶 → 𝐹 Fn 𝐴)
17	1, 16	syl 14	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
18		fvco2 5585	. . . 4 ⊢ (((inl ↾ 𝐴) Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
19	9, 18	sylan 283	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
20		fvres 5539	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
21	20	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
22	21	fveq2d 5519	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
23	3	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))))
24		fveq2 5515	. . . . . . . . 9 ⊢ (𝑥 = (inl‘𝑎) → (1st ‘𝑥) = (1st ‘(inl‘𝑎)))
25	24	eqeq1d 2186	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → ((1st ‘𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
26		fveq2 5515	. . . . . . . . 9 ⊢ (𝑥 = (inl‘𝑎) → (2nd ‘𝑥) = (2nd ‘(inl‘𝑎)))
27	26	fveq2d 5519	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐹‘(2nd ‘𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
28	26	fveq2d 5519	. . . . . . . 8 ⊢ (𝑥 = (inl‘𝑎) → (𝐺‘(2nd ‘𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
29	25, 27, 28	ifbieq12d 3560	. . . . . . 7 ⊢ (𝑥 = (inl‘𝑎) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
30	29	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
31		1stinl 7072	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → (1st ‘(inl‘𝑎)) = ∅)
32	31	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1st ‘(inl‘𝑎)) = ∅)
33	32	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → (1st ‘(inl‘𝑎)) = ∅)
34	33	iftrued 3541	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))) = (𝐹‘(2nd ‘(inl‘𝑎))))
35	30, 34	eqtrd 2210	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
36		djulcl 7049	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → (inl‘𝑎) ∈ (𝐴 ⊔ 𝐵))
37	36	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (inl‘𝑎) ∈ (𝐴 ⊔ 𝐵))
38	1	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐹:𝐴⟶𝐶)
39		2ndinl 7073	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐴 → (2nd ‘(inl‘𝑎)) = 𝑎)
40	39	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (2nd ‘(inl‘𝑎)) = 𝑎)
41		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
42	40, 41	eqeltrd 2254	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (2nd ‘(inl‘𝑎)) ∈ 𝐴)
43	38, 42	ffvelcdmd 5652	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) ∈ 𝐶)
44	23, 35, 37, 43	fvmptd 5597	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘(inl‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
45	22, 44	eqtrd 2210	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
46	40	fveq2d 5519	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹‘𝑎))
47	19, 45, 46	3eqtrd 2214	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐹‘𝑎))
48	15, 17, 47	eqfnfvd 5616	1 ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ⊆ wss 3129  ∅c0 3422  ifcif 3534   ↦ cmpt 4064  ran crn 4627   ↾ cres 4628   ∘ ccom 4630   Fn wfn 5211  ⟶wf 5212  –1-1→wf1 5213  ‘cfv 5216  1^st c1st 6138  2^nd c2nd 6139   ⊔ cdju 7035  inlcinl 7043

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-dju 7036  df-inl 7045  df-inr 7046

This theorem is referenced by:  updjud  7080