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Theorem bnj1400 31244
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 5472 . 2 dom 𝐴 = 𝑧𝐴 dom 𝑧
2 df-iun 4656 . . 3 𝑥𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥}
3 df-iun 4656 . . . 4 𝑧𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧𝐴 𝑦 ∈ dom 𝑧}
4 bnj1400.1 . . . . . . 7 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
54nfcii 2904 . . . . . 6 𝑥𝐴
6 nfcv 2913 . . . . . 6 𝑧𝐴
7 nfv 1995 . . . . . 6 𝑧 𝑦 ∈ dom 𝑥
8 nfv 1995 . . . . . 6 𝑥 𝑦 ∈ dom 𝑧
9 dmeq 5462 . . . . . . 7 (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
109eleq2d 2836 . . . . . 6 (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥𝑦 ∈ dom 𝑧))
115, 6, 7, 8, 10cbvrexf 3315 . . . . 5 (∃𝑥𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧𝐴 𝑦 ∈ dom 𝑧)
1211abbii 2888 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧𝐴 𝑦 ∈ dom 𝑧}
133, 12eqtr4i 2796 . . 3 𝑧𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥}
142, 13eqtr4i 2796 . 2 𝑥𝐴 dom 𝑥 = 𝑧𝐴 dom 𝑧
151, 14eqtr4i 2796 1 dom 𝐴 = 𝑥𝐴 dom 𝑥
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629   = wceq 1631  wcel 2145  {cab 2757  wrex 3062   cuni 4574   ciun 4654  dom cdm 5249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-dm 5259
This theorem is referenced by:  bnj1398  31440  bnj1450  31456  bnj1498  31467  bnj1501  31473
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