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Theorem bnj1400 31213
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 5489 . 2 dom 𝐴 = 𝑧𝐴 dom 𝑧
2 df-iun 4674 . . 3 𝑥𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥}
3 df-iun 4674 . . . 4 𝑧𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧𝐴 𝑦 ∈ dom 𝑧}
4 bnj1400.1 . . . . . . 7 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
54nfcii 2893 . . . . . 6 𝑥𝐴
6 nfcv 2902 . . . . . 6 𝑧𝐴
7 nfv 1992 . . . . . 6 𝑧 𝑦 ∈ dom 𝑥
8 nfv 1992 . . . . . 6 𝑥 𝑦 ∈ dom 𝑧
9 dmeq 5479 . . . . . . 7 (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧)
109eleq2d 2825 . . . . . 6 (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥𝑦 ∈ dom 𝑧))
115, 6, 7, 8, 10cbvrexf 3305 . . . . 5 (∃𝑥𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧𝐴 𝑦 ∈ dom 𝑧)
1211abbii 2877 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧𝐴 𝑦 ∈ dom 𝑧}
133, 12eqtr4i 2785 . . 3 𝑧𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ dom 𝑥}
142, 13eqtr4i 2785 . 2 𝑥𝐴 dom 𝑥 = 𝑧𝐴 dom 𝑧
151, 14eqtr4i 2785 1 dom 𝐴 = 𝑥𝐴 dom 𝑥
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630   = wceq 1632  wcel 2139  {cab 2746  wrex 3051   cuni 4588   ciun 4672  dom cdm 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-dm 5276
This theorem is referenced by:  bnj1398  31409  bnj1450  31425  bnj1498  31436  bnj1501  31442
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