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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 6843 | . . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅)) | |
2 | reseq2 5933 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = (𝐹 ↾ ∅)) | |
3 | res0 5942 | . . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅ | |
4 | 2, 3 | eqtrdi 2793 | . . . . 5 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = ∅) |
5 | 4 | fveq2d 6847 | . . . 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 3535 | . 2 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅)) |
9 | 0ex 5265 | . . 3 ⊢ ∅ ∈ V | |
10 | iftrue 4493 | . . . 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 6949 | . . 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 1542 ∈ wcel 2107 Vcvv 3446 ∅c0 4283 ifcif 4487 ∪ cuni 4866 ↦ cmpt 5189 dom cdm 5634 ran crn 5635 ↾ cres 5636 Lim wlim 6319 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 |
This theorem is referenced by: rdg0 8368 |
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