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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 6902 | . . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅)) | |
2 | reseq2 5984 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = (𝐹 ↾ ∅)) | |
3 | res0 5993 | . . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅ | |
4 | 2, 3 | eqtrdi 2784 | . . . . 5 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = ∅) |
5 | 4 | fveq2d 6906 | . . . 4 ⊢ (𝑦 = ∅ → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘∅)) |
6 | 1, 5 | eqeq12d 2744 | . . 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 3565 | . 2 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅)) |
9 | 0ex 5311 | . . 3 ⊢ ∅ ∈ V | |
10 | iftrue 4538 | . . . 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 7010 | . . 3 ⊢ (∅ ∈ V → (𝐺‘∅) = 𝐴) |
14 | 9, 13 | ax-mp 5 | . 2 ⊢ (𝐺‘∅) = 𝐴 |
15 | 8, 14 | eqtrdi 2784 | 1 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∅c0 4326 ifcif 4532 ∪ cuni 4912 ↦ cmpt 5235 dom cdm 5682 ran crn 5683 ↾ cres 5684 Lim wlim 6375 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-res 5694 df-iota 6505 df-fun 6555 df-fv 6561 |
This theorem is referenced by: rdg0 8450 |
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