![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6885 | . . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅)) | |
2 | reseq2 5970 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = (𝐹 ↾ ∅)) | |
3 | res0 5979 | . . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅ | |
4 | 2, 3 | eqtrdi 2782 | . . . . 5 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = ∅) |
5 | 4 | fveq2d 6889 | . . . 4 ⊢ (𝑦 = ∅ → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘∅)) |
6 | 1, 5 | eqeq12d 2742 | . . 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 3560 | . 2 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅)) |
9 | 0ex 5300 | . . 3 ⊢ ∅ ∈ V | |
10 | iftrue 4529 | . . . 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 6992 | . . 3 ⊢ (∅ ∈ V → (𝐺‘∅) = 𝐴) |
14 | 9, 13 | ax-mp 5 | . 2 ⊢ (𝐺‘∅) = 𝐴 |
15 | 8, 14 | eqtrdi 2782 | 1 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∅c0 4317 ifcif 4523 ∪ cuni 4902 ↦ cmpt 5224 dom cdm 5669 ran crn 5670 ↾ cres 5671 Lim wlim 6359 ‘cfv 6537 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6489 df-fun 6539 df-fv 6545 |
This theorem is referenced by: rdg0 8422 |
Copyright terms: Public domain | W3C validator |