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Theorem rexrnmpt 5223
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
ralrnmpt.1 𝐹 = (x AB)
ralrnmpt.2 (y = B → (ψχ))
Assertion
Ref Expression
rexrnmpt (x A B 𝑉 → (y ran 𝐹ψx A χ))
Distinct variable groups:   x,A   y,B   χ,y   y,𝐹   ψ,x
Allowed substitution hints:   ψ(y)   χ(x)   A(y)   B(x)   𝐹(x)   𝑉(x,y)

Proof of Theorem rexrnmpt
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ralrnmpt.1 . . . . 5 𝐹 = (x AB)
21fnmpt 4939 . . . 4 (x A B 𝑉𝐹 Fn A)
3 dfsbcq 2734 . . . . 5 (w = (𝐹z) → ([w / y]ψ[(𝐹z) / y]ψ))
43rexrn 5217 . . . 4 (𝐹 Fn A → (w ran 𝐹[w / y]ψz A [(𝐹z) / y]ψ))
52, 4syl 14 . . 3 (x A B 𝑉 → (w ran 𝐹[w / y]ψz A [(𝐹z) / y]ψ))
6 nfv 1394 . . . . 5 wψ
7 nfsbc1v 2750 . . . . 5 y[w / y]ψ
8 sbceq1a 2741 . . . . 5 (y = w → (ψ[w / y]ψ))
96, 7, 8cbvrex 2499 . . . 4 (y ran 𝐹ψw ran 𝐹[w / y]ψ)
109bicomi 123 . . 3 (w ran 𝐹[w / y]ψy ran 𝐹ψ)
11 nfmpt1 3813 . . . . . . 7 x(x AB)
121, 11nfcxfr 2148 . . . . . 6 x𝐹
13 nfcv 2151 . . . . . 6 xz
1412, 13nffv 5098 . . . . 5 x(𝐹z)
15 nfv 1394 . . . . 5 xψ
1614, 15nfsbc 2752 . . . 4 x[(𝐹z) / y]ψ
17 nfv 1394 . . . 4 z[(𝐹x) / y]ψ
18 fveq2 5091 . . . . 5 (z = x → (𝐹z) = (𝐹x))
1918sbceq1d 2737 . . . 4 (z = x → ([(𝐹z) / y]ψ[(𝐹x) / y]ψ))
2016, 17, 19cbvrex 2499 . . 3 (z A [(𝐹z) / y]ψx A [(𝐹x) / y]ψ)
215, 10, 203bitr3g 211 . 2 (x A B 𝑉 → (y ran 𝐹ψx A [(𝐹x) / y]ψ))
221fvmpt2 5167 . . . . . 6 ((x A B 𝑉) → (𝐹x) = B)
2322sbceq1d 2737 . . . . 5 ((x A B 𝑉) → ([(𝐹x) / y]ψ[B / y]ψ))
24 ralrnmpt.2 . . . . . . 7 (y = B → (ψχ))
2524sbcieg 2763 . . . . . 6 (B 𝑉 → ([B / y]ψχ))
2625adantl 262 . . . . 5 ((x A B 𝑉) → ([B / y]ψχ))
2723, 26bitrd 177 . . . 4 ((x A B 𝑉) → ([(𝐹x) / y]ψχ))
2827ralimiaa 2352 . . 3 (x A B 𝑉x A ([(𝐹x) / y]ψχ))
29 pm5.32 426 . . . . . 6 ((x A → ([(𝐹x) / y]ψχ)) ↔ ((x A [(𝐹x) / y]ψ) ↔ (x A χ)))
3029albii 1332 . . . . 5 (x(x A → ([(𝐹x) / y]ψχ)) ↔ x((x A [(𝐹x) / y]ψ) ↔ (x A χ)))
31 exbi 1468 . . . . 5 (x((x A [(𝐹x) / y]ψ) ↔ (x A χ)) → (x(x A [(𝐹x) / y]ψ) ↔ x(x A χ)))
3230, 31sylbi 114 . . . 4 (x(x A → ([(𝐹x) / y]ψχ)) → (x(x A [(𝐹x) / y]ψ) ↔ x(x A χ)))
33 df-ral 2280 . . . 4 (x A ([(𝐹x) / y]ψχ) ↔ x(x A → ([(𝐹x) / y]ψχ)))
34 df-rex 2281 . . . . 5 (x A [(𝐹x) / y]ψx(x A [(𝐹x) / y]ψ))
35 df-rex 2281 . . . . 5 (x A χx(x A χ))
3634, 35bibi12i 218 . . . 4 ((x A [(𝐹x) / y]ψx A χ) ↔ (x(x A [(𝐹x) / y]ψ) ↔ x(x A χ)))
3732, 33, 363imtr4i 190 . . 3 (x A ([(𝐹x) / y]ψχ) → (x A [(𝐹x) / y]ψx A χ))
3828, 37syl 14 . 2 (x A B 𝑉 → (x A [(𝐹x) / y]ψx A χ))
3921, 38bitrd 177 1 (x A B 𝑉 → (y ran 𝐹ψx A χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1221   = wceq 1223  wex 1354   wcel 1366  wral 2275  wrex 2276  [wsbc 2732  cmpt 3781  ran crn 4261   Fn wfn 4812  cfv 4817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-sbc 2733  df-csb 2821  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-iota 4782  df-fun 4819  df-fn 4820  df-fv 4825
This theorem is referenced by: (None)
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