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Mirrors > Home > MPE Home > Th. List > tz7.44lem1 | Structured version Visualization version GIF version |
Description: 𝐺 is a function. Lemma for tz7.44-1 7901, tz7.44-2 7902, and tz7.44-3 7903. (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 6267 | . . 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 6558 | . . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V | |
3 | vex 3443 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | rnexg 7477 | . . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V) | |
5 | uniexg 7332 | . . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V) | |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ∪ ran 𝑥 ∈ V |
7 | nlim0 6131 | . . . . . 6 ⊢ ¬ Lim ∅ | |
8 | dm0 5683 | . . . . . . 7 ⊢ dom ∅ = ∅ | |
9 | limeq 6085 | . . . . . . 7 ⊢ (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (Lim dom ∅ ↔ Lim ∅) |
11 | 7, 10 | mtbir 324 | . . . . 5 ⊢ ¬ Lim dom ∅ |
12 | dmeq 5665 | . . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅) | |
13 | limeq 6085 | . . . . . . 7 ⊢ (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) | |
14 | 12, 13 | syl 17 | . . . . . 6 ⊢ (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) |
15 | 14 | biimpa 477 | . . . . 5 ⊢ ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅) |
16 | 11, 15 | mto 198 | . . . 4 ⊢ ¬ (𝑥 = ∅ ∧ Lim dom 𝑥) |
17 | 2, 6, 16 | moeq3 3644 | . . 3 ⊢ ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥)) |
18 | 1, 17 | mpgbir 1785 | . 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 6253 | . 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 232 | 1 ⊢ Fun 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∨ wo 842 ∨ w3o 1079 = wceq 1525 ∈ wcel 2083 ∃*wmo 2576 Vcvv 3440 ∅c0 4217 ∪ cuni 4751 {copab 5030 dom cdm 5450 ran crn 5451 Lim wlim 6074 Fun wfun 6226 ‘cfv 6232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-ord 6076 df-lim 6078 df-iota 6196 df-fun 6234 df-fv 6240 |
This theorem is referenced by: (None) |
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