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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1400 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1400.1 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Ref | Expression |
---|---|
bnj1400 | ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmuni 5927 | . 2 ⊢ dom ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 dom 𝑧 | |
2 | df-iun 4997 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} | |
3 | df-iun 4997 | . . . 4 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧} | |
4 | bnj1400.1 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
5 | 4 | nfcii 2891 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 |
6 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
7 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑧 𝑦 ∈ dom 𝑥 | |
8 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ dom 𝑧 | |
9 | dmeq 5916 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧) | |
10 | 9 | eleq2d 2824 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥 ↔ 𝑦 ∈ dom 𝑧)) |
11 | 5, 6, 7, 8, 10 | cbvrexfw 3302 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧) |
12 | 11 | abbii 2806 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧} |
13 | 3, 12 | eqtr4i 2765 | . . 3 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} |
14 | 2, 13 | eqtr4i 2765 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = ∪ 𝑧 ∈ 𝐴 dom 𝑧 |
15 | 1, 14 | eqtr4i 2765 | 1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 = wceq 1536 ∈ wcel 2105 {cab 2711 ∃wrex 3067 ∪ cuni 4911 ∪ ciun 4995 dom cdm 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-dm 5698 |
This theorem is referenced by: bnj1398 35026 bnj1450 35042 bnj1498 35053 bnj1501 35059 |
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