![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tz7.44lem1 | Structured version Visualization version GIF version |
Description: The ordered pair abstraction 𝐺 defined in the hypothesis is a function. This was a lemma for tz7.44-1 8353, tz7.44-2 8354, and tz7.44-3 8355 when they used that definition of 𝐺. Now, they use the maps-to df-mpt 5190 idiom so this lemma is not needed anymore, but is kept in case other applications (for instance in intuitionistic set theory) need it. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
tz7.44lem1.1 | ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} |
Ref | Expression |
---|---|
tz7.44lem1 | ⊢ Fun 𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funopab 6537 | . . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} ↔ ∀𝑥∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))) | |
2 | fvex 6856 | . . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V | |
3 | vex 3450 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | rnexg 7842 | . . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V) | |
5 | uniexg 7678 | . . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V) | |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ∪ ran 𝑥 ∈ V |
7 | nlim0 6377 | . . . . . 6 ⊢ ¬ Lim ∅ | |
8 | dm0 5877 | . . . . . . 7 ⊢ dom ∅ = ∅ | |
9 | limeq 6330 | . . . . . . 7 ⊢ (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (Lim dom ∅ ↔ Lim ∅) |
11 | 7, 10 | mtbir 323 | . . . . 5 ⊢ ¬ Lim dom ∅ |
12 | dmeq 5860 | . . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅) | |
13 | limeq 6330 | . . . . . . 7 ⊢ (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) | |
14 | 12, 13 | syl 17 | . . . . . 6 ⊢ (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) |
15 | 14 | biimpa 478 | . . . . 5 ⊢ ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅) |
16 | 11, 15 | mto 196 | . . . 4 ⊢ ¬ (𝑥 = ∅ ∧ Lim dom 𝑥) |
17 | 2, 6, 16 | moeq3 3671 | . . 3 ⊢ ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥)) |
18 | 1, 17 | mpgbir 1802 | . 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} |
19 | tz7.44lem1.1 | . . 3 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} | |
20 | 19 | funeqi 6523 | . 2 ⊢ (Fun 𝐺 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))}) |
21 | 18, 20 | mpbir 230 | 1 ⊢ Fun 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ∃*wmo 2537 Vcvv 3446 ∅c0 4283 ∪ cuni 4866 {copab 5168 dom cdm 5634 ran crn 5635 Lim wlim 6319 Fun wfun 6491 ‘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 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-ne 2945 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-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-ord 6321 df-lim 6323 df-iota 6449 df-fun 6499 df-fv 6505 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |