![]() |
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 6907 | . . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅)) | |
2 | reseq2 5995 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = (𝐹 ↾ ∅)) | |
3 | res0 6004 | . . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅ | |
4 | 2, 3 | eqtrdi 2791 | . . . . 5 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = ∅) |
5 | 4 | fveq2d 6911 | . . . 4 ⊢ (𝑦 = ∅ → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘∅)) |
6 | 1, 5 | eqeq12d 2751 | . . 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 5313 | . . 3 ⊢ ∅ ∈ V | |
10 | iftrue 4537 | . . . 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 2791 | 1 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ifcif 4531 ∪ cuni 4912 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 ↾ cres 5691 Lim wlim 6387 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 |
This theorem is referenced by: rdg0 8460 |
Copyright terms: Public domain | W3C validator |