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Theorem bnj1400 30635
 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 5296 . 2 dom 𝐴 = 𝑧𝐴 dom 𝑧
2 df-iun 4489 . . 3 𝑥𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥}
3 df-iun 4489 . . . 4 𝑧𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧𝐴 𝑦 ∈ dom 𝑧}
4 bnj1400.1 . . . . . . 7 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
54nfcii 2752 . . . . . 6 𝑥𝐴
6 nfcv 2761 . . . . . 6 𝑧𝐴
7 nfv 1840 . . . . . 6 𝑧 𝑦 ∈ dom 𝑥
8 nfv 1840 . . . . . 6 𝑥 𝑦 ∈ dom 𝑧
9 dmeq 5286 . . . . . . 7 (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
109eleq2d 2684 . . . . . 6 (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥𝑦 ∈ dom 𝑧))
115, 6, 7, 8, 10cbvrexf 3154 . . . . 5 (∃𝑥𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧𝐴 𝑦 ∈ dom 𝑧)
1211abbii 2736 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧𝐴 𝑦 ∈ dom 𝑧}
133, 12eqtr4i 2646 . . 3 𝑧𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥}
142, 13eqtr4i 2646 . 2 𝑥𝐴 dom 𝑥 = 𝑧𝐴 dom 𝑧
151, 14eqtr4i 2646 1 dom 𝐴 = 𝑥𝐴 dom 𝑥
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478   = wceq 1480   ∈ wcel 1987  {cab 2607  ∃wrex 2908  ∪ cuni 4404  ∪ ciun 4487  dom cdm 5076 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-dm 5086 This theorem is referenced by:  bnj1398  30831  bnj1450  30847  bnj1498  30858  bnj1501  30864
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