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| Mirrors > Home > MPE Home > Th. List > Mathboxes > functhinclem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for functhinc 49097. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| functhinclem3.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| functhinclem3.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| functhinclem3.m | ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌)) | 
| functhinclem3.g | ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))) | 
| functhinclem3.1 | ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅)) | 
| functhinclem3.2 | ⊢ (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) | 
| Ref | Expression | 
|---|---|
| functhinclem3 | ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | functhinclem3.g | . . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))) | |
| 2 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋) | |
| 3 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
| 4 | 2, 3 | oveq12d 7449 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) | 
| 5 | 2 | fveq2d 6910 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑥) = (𝐹‘𝑋)) | 
| 6 | 3 | fveq2d 6910 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑌)) | 
| 7 | 5, 6 | oveq12d 7449 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) | 
| 8 | 4, 7 | xpeq12d 5716 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))) | 
| 9 | functhinclem3.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 10 | functhinclem3.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 11 | ovex 7464 | . . . . . 6 ⊢ (𝑋𝐻𝑌) ∈ V | |
| 12 | ovex 7464 | . . . . . 6 ⊢ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∈ V | |
| 13 | 11, 12 | xpex 7773 | . . . . 5 ⊢ ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) ∈ V | 
| 14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) ∈ V) | 
| 15 | 1, 8, 9, 10, 14 | ovmpod 7585 | . . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))) | 
| 16 | eqid 2737 | . . . 4 ⊢ ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) | |
| 17 | functhinclem3.1 | . . . 4 ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅)) | |
| 18 | functhinclem3.2 | . . . 4 ⊢ (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) | |
| 19 | 16, 17, 18 | mofeu 48757 | . . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↔ (𝑋𝐺𝑌) = ((𝑋𝐻𝑌) × ((𝐹‘𝑋)𝐽(𝐹‘𝑌))))) | 
| 20 | 15, 19 | mpbird 257 | . 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌))) | 
| 21 | functhinclem3.m | . 2 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌)) | |
| 22 | 20, 21 | ffvelcdmd 7105 | 1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) | 
| 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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