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Theorem updjudhcoinlf 7073
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 (𝜑𝐹:𝐴𝐶)
updjud.g (𝜑𝐺:𝐵𝐶)
updjudhf.h 𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
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.
StepHypRef Expression
1 updjud.f . . . . 5 (𝜑𝐹:𝐴𝐶)
2 updjud.g . . . . 5 (𝜑𝐺:𝐵𝐶)
3 updjudhf.h . . . . 5 𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
41, 2, 3updjudhf 7072 . . . 4 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
5 ffn 5361 . . . 4 (𝐻:(𝐴𝐵)⟶𝐶𝐻 Fn (𝐴𝐵))
64, 5syl 14 . . 3 (𝜑𝐻 Fn (𝐴𝐵))
7 inlresf1 7054 . . . 4 (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)
8 f1fn 5419 . . . 4 ((inl ↾ 𝐴):𝐴1-1→(𝐴𝐵) → (inl ↾ 𝐴) Fn 𝐴)
97, 8mp1i 10 . . 3 (𝜑 → (inl ↾ 𝐴) Fn 𝐴)
10 f1f 5417 . . . . 5 ((inl ↾ 𝐴):𝐴1-1→(𝐴𝐵) → (inl ↾ 𝐴):𝐴⟶(𝐴𝐵))
117, 10ax-mp 5 . . . 4 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
12 frn 5370 . . . 4 ((inl ↾ 𝐴):𝐴⟶(𝐴𝐵) → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
1311, 12mp1i 10 . . 3 (𝜑 → ran (inl ↾ 𝐴) ⊆ (𝐴𝐵))
14 fnco 5320 . . 3 ((𝐻 Fn (𝐴𝐵) ∧ (inl ↾ 𝐴) Fn 𝐴 ∧ ran (inl ↾ 𝐴) ⊆ (𝐴𝐵)) → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
156, 9, 13, 14syl3anc 1238 . 2 (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) Fn 𝐴)
16 ffn 5361 . . 3 (𝐹:𝐴𝐶𝐹 Fn 𝐴)
171, 16syl 14 . 2 (𝜑𝐹 Fn 𝐴)
18 fvco2 5581 . . . 4 (((inl ↾ 𝐴) Fn 𝐴𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
199, 18sylan 283 . . 3 ((𝜑𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐻‘((inl ↾ 𝐴)‘𝑎)))
20 fvres 5535 . . . . . 6 (𝑎𝐴 → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
2120adantl 277 . . . . 5 ((𝜑𝑎𝐴) → ((inl ↾ 𝐴)‘𝑎) = (inl‘𝑎))
2221fveq2d 5515 . . . 4 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐻‘(inl‘𝑎)))
233a1i 9 . . . . 5 ((𝜑𝑎𝐴) → 𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥)))))
24 fveq2 5511 . . . . . . . . 9 (𝑥 = (inl‘𝑎) → (1st𝑥) = (1st ‘(inl‘𝑎)))
2524eqeq1d 2186 . . . . . . . 8 (𝑥 = (inl‘𝑎) → ((1st𝑥) = ∅ ↔ (1st ‘(inl‘𝑎)) = ∅))
26 fveq2 5511 . . . . . . . . 9 (𝑥 = (inl‘𝑎) → (2nd𝑥) = (2nd ‘(inl‘𝑎)))
2726fveq2d 5515 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐹‘(2nd𝑥)) = (𝐹‘(2nd ‘(inl‘𝑎))))
2826fveq2d 5515 . . . . . . . 8 (𝑥 = (inl‘𝑎) → (𝐺‘(2nd𝑥)) = (𝐺‘(2nd ‘(inl‘𝑎))))
2925, 27, 28ifbieq12d 3560 . . . . . . 7 (𝑥 = (inl‘𝑎) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
3029adantl 277 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))))
31 1stinl 7067 . . . . . . . . 9 (𝑎𝐴 → (1st ‘(inl‘𝑎)) = ∅)
3231adantl 277 . . . . . . . 8 ((𝜑𝑎𝐴) → (1st ‘(inl‘𝑎)) = ∅)
3332adantr 276 . . . . . . 7 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → (1st ‘(inl‘𝑎)) = ∅)
3433iftrued 3541 . . . . . 6 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st ‘(inl‘𝑎)) = ∅, (𝐹‘(2nd ‘(inl‘𝑎))), (𝐺‘(2nd ‘(inl‘𝑎)))) = (𝐹‘(2nd ‘(inl‘𝑎))))
3530, 34eqtrd 2210 . . . . 5 (((𝜑𝑎𝐴) ∧ 𝑥 = (inl‘𝑎)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) = (𝐹‘(2nd ‘(inl‘𝑎))))
36 djulcl 7044 . . . . . 6 (𝑎𝐴 → (inl‘𝑎) ∈ (𝐴𝐵))
3736adantl 277 . . . . 5 ((𝜑𝑎𝐴) → (inl‘𝑎) ∈ (𝐴𝐵))
381adantr 276 . . . . . 6 ((𝜑𝑎𝐴) → 𝐹:𝐴𝐶)
39 2ndinl 7068 . . . . . . . 8 (𝑎𝐴 → (2nd ‘(inl‘𝑎)) = 𝑎)
4039adantl 277 . . . . . . 7 ((𝜑𝑎𝐴) → (2nd ‘(inl‘𝑎)) = 𝑎)
41 simpr 110 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎𝐴)
4240, 41eqeltrd 2254 . . . . . 6 ((𝜑𝑎𝐴) → (2nd ‘(inl‘𝑎)) ∈ 𝐴)
4338, 42ffvelcdmd 5648 . . . . 5 ((𝜑𝑎𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) ∈ 𝐶)
4423, 35, 37, 43fvmptd 5593 . . . 4 ((𝜑𝑎𝐴) → (𝐻‘(inl‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
4522, 44eqtrd 2210 . . 3 ((𝜑𝑎𝐴) → (𝐻‘((inl ↾ 𝐴)‘𝑎)) = (𝐹‘(2nd ‘(inl‘𝑎))))
4640fveq2d 5515 . . 3 ((𝜑𝑎𝐴) → (𝐹‘(2nd ‘(inl‘𝑎))) = (𝐹𝑎))
4719, 45, 463eqtrd 2214 . 2 ((𝜑𝑎𝐴) → ((𝐻 ∘ (inl ↾ 𝐴))‘𝑎) = (𝐹𝑎))
4815, 17, 47eqfnfvd 5612 1 (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wss 3129  c0 3422  ifcif 3534  cmpt 4061  ran crn 4624  cres 4625  ccom 4627   Fn wfn 5207  wf 5208  1-1wf1 5209  cfv 5212  1st c1st 6133  2nd c2nd 6134  cdju 7030  inlcinl 7038
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 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-1o 6411  df-dju 7031  df-inl 7040  df-inr 7041
This theorem is referenced by:  updjud  7075
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