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Theorem fnopabg 5205
 Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Hypothesis
Ref Expression
fnopabg.1 F = {x, y (x A φ)}
Assertion
Ref Expression
fnopabg (x A ∃!yφF Fn A)
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)   F(x,y)

Proof of Theorem fnopabg
StepHypRef Expression
1 moanimv 2262 . . . . . 6 (∃*y(x A φ) ↔ (x A∃*yφ))
21albii 1566 . . . . 5 (x∃*y(x A φ) ↔ x(x A∃*yφ))
3 funopab 5139 . . . . 5 (Fun {x, y (x A φ)} ↔ x∃*y(x A φ))
4 df-ral 2619 . . . . 5 (x A ∃*yφx(x A∃*yφ))
52, 3, 43bitr4ri 269 . . . 4 (x A ∃*yφ ↔ Fun {x, y (x A φ)})
6 dmopab3 4917 . . . 4 (x A yφ ↔ dom {x, y (x A φ)} = A)
75, 6anbi12i 678 . . 3 ((x A ∃*yφ x A yφ) ↔ (Fun {x, y (x A φ)} dom {x, y (x A φ)} = A))
8 r19.26 2746 . . 3 (x A (∃*yφ yφ) ↔ (x A ∃*yφ x A yφ))
9 df-fn 4790 . . 3 ({x, y (x A φ)} Fn A ↔ (Fun {x, y (x A φ)} dom {x, y (x A φ)} = A))
107, 8, 93bitr4i 268 . 2 (x A (∃*yφ yφ) ↔ {x, y (x A φ)} Fn A)
11 eu5 2242 . . . 4 (∃!yφ ↔ (yφ ∃*yφ))
12 ancom 437 . . . 4 ((yφ ∃*yφ) ↔ (∃*yφ yφ))
1311, 12bitri 240 . . 3 (∃!yφ ↔ (∃*yφ yφ))
1413ralbii 2638 . 2 (x A ∃!yφx A (∃*yφ yφ))
15 fnopabg.1 . . 3 F = {x, y (x A φ)}
1615fneq1i 5178 . 2 (F Fn A ↔ {x, y (x A φ)} Fn A)
1710, 14, 163bitr4i 268 1 (x A ∃!yφF Fn A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∀wral 2614  {copab 4622  dom cdm 4772  Fun wfun 4775   Fn wfn 4776 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790 This theorem is referenced by:  fnopab2g  5206  fnopab  5207
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