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| Mirrors > Home > MPE Home > Th. List > tz7.44-1 | Structured version Visualization version GIF version | ||
| Description: The value of 𝐹 at ∅. Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| tz7.44.1 | ⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥))))) | 
| tz7.44.2 | ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦))) | 
| tz7.44-1.3 | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| tz7.44-1 | ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6906 | . . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅)) | |
| 2 | reseq2 5992 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = (𝐹 ↾ ∅)) | |
| 3 | res0 6001 | . . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅ | |
| 4 | 2, 3 | eqtrdi 2793 | . . . . 5 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = ∅) | 
| 5 | 4 | fveq2d 6910 | . . . 4 ⊢ (𝑦 = ∅ → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘∅)) | 
| 6 | 1, 5 | eqeq12d 2753 | . . 3 ⊢ (𝑦 = ∅ → ((𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)) ↔ (𝐹‘∅) = (𝐺‘∅))) | 
| 7 | tz7.44.2 | . . 3 ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦))) | |
| 8 | 6, 7 | vtoclga 3577 | . 2 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅)) | 
| 9 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 10 | iftrue 4531 | . . . 4 ⊢ (𝑥 = ∅ → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))) = 𝐴) | |
| 11 | tz7.44.1 | . . . 4 ⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥))))) | |
| 12 | tz7.44-1.3 | . . . 4 ⊢ 𝐴 ∈ V | |
| 13 | 10, 11, 12 | fvmpt 7016 | . . 3 ⊢ (∅ ∈ V → (𝐺‘∅) = 𝐴) | 
| 14 | 9, 13 | ax-mp 5 | . 2 ⊢ (𝐺‘∅) = 𝐴 | 
| 15 | 8, 14 | eqtrdi 2793 | 1 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ifcif 4525 ∪ cuni 4907 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 ↾ cres 5687 Lim wlim 6385 ‘cfv 6561 | 
| 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-pr 5432 | 
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 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-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 | 
| This theorem is referenced by: rdg0 8461 | 
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