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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 5334 | . 2 ⊢ dom ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 dom 𝑧 | |
2 | df-iun 4522 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} | |
3 | df-iun 4522 | . . . 4 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧} | |
4 | bnj1400.1 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
5 | 4 | nfcii 2755 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 |
6 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
7 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑧 𝑦 ∈ dom 𝑥 | |
8 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ dom 𝑧 | |
9 | dmeq 5324 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧) | |
10 | 9 | eleq2d 2687 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥 ↔ 𝑦 ∈ dom 𝑧)) |
11 | 5, 6, 7, 8, 10 | cbvrexf 3166 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧) |
12 | 11 | abbii 2739 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧} |
13 | 3, 12 | eqtr4i 2647 | . . 3 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} |
14 | 2, 13 | eqtr4i 2647 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = ∪ 𝑧 ∈ 𝐴 dom 𝑧 |
15 | 1, 14 | eqtr4i 2647 | 1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 ∪ cuni 4436 ∪ ciun 4520 dom cdm 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-dm 5124 |
This theorem is referenced by: bnj1398 31102 bnj1450 31118 bnj1498 31129 bnj1501 31135 |
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