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Theorem functhinclem3 47216
Description: Lemma for functhinc 47218. 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 769 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
3 simprr 771 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
42, 3oveq12d 7395 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
52fveq2d 6866 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝐹𝑥) = (𝐹𝑋))
63fveq2d 6866 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝐹𝑦) = (𝐹𝑌))
75, 6oveq12d 7395 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝑋)𝐽(𝐹𝑌)))
84, 7xpeq12d 5684 . . . 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 7410 . . . . . 6 (𝑋𝐻𝑌) ∈ V
12 ovex 7410 . . . . . 6 ((𝐹𝑋)𝐽(𝐹𝑌)) ∈ V
1311, 12xpex 7707 . . . . 5 ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌))) ∈ V
1413a1i 11 . . . 4 (𝜑 → ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌))) ∈ V)
151, 8, 9, 10, 14ovmpod 7527 . . 3 (𝜑 → (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌))))
16 eqid 2731 . . . 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 47067 . . 3 (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)) ↔ (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹𝑋)𝐽(𝐹𝑌)))))
2015, 19mpbird 256 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)))
21 functhinclem3.m . 2 (𝜑𝑀 ∈ (𝑋𝐻𝑌))
2220, 21ffvelcdmd 7056 1 (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ∃*wmo 2531  Vcvv 3459  c0 4302   × cxp 5651  wf 6512  cfv 6516  (class class class)co 7377  cmpo 7379
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 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3364  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382
This theorem is referenced by:  functhinclem4  47217
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