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Theorem functhinclem3 49095
Description: Lemma for functhinc 49097. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)
Hypotheses
Ref Expression
functhinclem3.x (𝜑𝑋𝐵)
functhinclem3.y (𝜑𝑌𝐵)
functhinclem3.m (𝜑𝑀 ∈ (𝑋𝐻𝑌))
functhinclem3.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦)))))
functhinclem3.1 (𝜑 → (((𝐹𝑋)𝐽(𝐹𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
functhinclem3.2 (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Assertion
Ref Expression
functhinclem3 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Distinct variable groups:   𝑛,𝐹   𝑥,𝐹,𝑦   𝑥,𝐻,𝑦   𝑛,𝐽   𝑥,𝐽,𝑦   𝑛,𝑋   𝑥,𝑋,𝑦   𝑛,𝑌   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑛)   𝐵(𝑥,𝑦,𝑛)   𝐺(𝑥,𝑦,𝑛)   𝐻(𝑛)   𝑀(𝑥,𝑦,𝑛)

Proof of Theorem functhinclem3
StepHypRef Expression
1 functhinclem3.g . . . 4 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦)))))
2 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
3 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
42, 3oveq12d 7449 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
52fveq2d 6910 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝐹𝑥) = (𝐹𝑋))
63fveq2d 6910 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝐹𝑦) = (𝐹𝑌))
75, 6oveq12d 7449 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝑋)𝐽(𝐹𝑌)))
84, 7xpeq12d 5716 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑥𝐻𝑦) × ((𝐹𝑥)𝐽(𝐹𝑦))) = ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌))))
9 functhinclem3.x . . . 4 (𝜑𝑋𝐵)
10 functhinclem3.y . . . 4 (𝜑𝑌𝐵)
11 ovex 7464 . . . . . 6 (𝑋𝐻𝑌) ∈ V
12 ovex 7464 . . . . . 6 ((𝐹𝑋)𝐽(𝐹𝑌)) ∈ V
1311, 12xpex 7773 . . . . 5 ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌))) ∈ V
1413a1i 11 . . . 4 (𝜑 → ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌))) ∈ V)
151, 8, 9, 10, 14ovmpod 7585 . . 3 (𝜑 → (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌))))
16 eqid 2737 . . . 4 ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌))) = ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌)))
17 functhinclem3.1 . . . 4 (𝜑 → (((𝐹𝑋)𝐽(𝐹𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
18 functhinclem3.2 . . . 4 (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
1916, 17, 18mofeu 48757 . . 3 (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)) ↔ (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌)))))
2015, 19mpbird 257 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)))
21 functhinclem3.m . 2 (𝜑𝑀 ∈ (𝑋𝐻𝑌))
2220, 21ffvelcdmd 7105 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  ∃*wmo 2538  Vcvv 3480  c0 4333   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  functhinclem4  49096
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