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Theorem bnj1400 34827
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1400.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Assertion
Ref Expression
bnj1400 dom 𝐴 = 𝑥𝐴 dom 𝑥
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bnj1400
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dmuni 5927 . 2 dom 𝐴 = 𝑧𝐴 dom 𝑧
2 df-iun 4997 . . 3 𝑥𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥}
3 df-iun 4997 . . . 4 𝑧𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧𝐴 𝑦 ∈ dom 𝑧}
4 bnj1400.1 . . . . . . 7 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
54nfcii 2891 . . . . . 6 𝑥𝐴
6 nfcv 2902 . . . . . 6 𝑧𝐴
7 nfv 1911 . . . . . 6 𝑧 𝑦 ∈ dom 𝑥
8 nfv 1911 . . . . . 6 𝑥 𝑦 ∈ dom 𝑧
9 dmeq 5916 . . . . . . 7 (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
109eleq2d 2824 . . . . . 6 (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥𝑦 ∈ dom 𝑧))
115, 6, 7, 8, 10cbvrexfw 3302 . . . . 5 (∃𝑥𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧𝐴 𝑦 ∈ dom 𝑧)
1211abbii 2806 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧𝐴 𝑦 ∈ dom 𝑧}
133, 12eqtr4i 2765 . . 3 𝑧𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥}
142, 13eqtr4i 2765 . 2 𝑥𝐴 dom 𝑥 = 𝑧𝐴 dom 𝑧
151, 14eqtr4i 2765 1 dom 𝐴 = 𝑥𝐴 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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