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Theorem rexrnmpt 5253
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 4968 . . . 4 (x A B 𝑉𝐹 Fn A)
3 dfsbcq 2760 . . . . 5 (w = (𝐹z) → ([w / y]ψ[(𝐹z) / y]ψ))
43rexrn 5247 . . . 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 1418 . . . . 5 wψ
7 nfsbc1v 2776 . . . . 5 y[w / y]ψ
8 sbceq1a 2767 . . . . 5 (y = w → (ψ[w / y]ψ))
96, 7, 8cbvrex 2524 . . . 4 (y ran 𝐹ψw ran 𝐹[w / y]ψ)
109bicomi 123 . . 3 (w ran 𝐹[w / y]ψy ran 𝐹ψ)
11 nfmpt1 3841 . . . . . . 7 x(x AB)
121, 11nfcxfr 2172 . . . . . 6 x𝐹
13 nfcv 2175 . . . . . 6 xz
1412, 13nffv 5128 . . . . 5 x(𝐹z)
15 nfv 1418 . . . . 5 xψ
1614, 15nfsbc 2778 . . . 4 x[(𝐹z) / y]ψ
17 nfv 1418 . . . 4 z[(𝐹x) / y]ψ
18 fveq2 5121 . . . . 5 (z = x → (𝐹z) = (𝐹x))
1918sbceq1d 2763 . . . 4 (z = x → ([(𝐹z) / y]ψ[(𝐹x) / y]ψ))
2016, 17, 19cbvrex 2524 . . 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 5197 . . . . . 6 ((x A B 𝑉) → (𝐹x) = B)
2322sbceq1d 2763 . . . . 5 ((x A B 𝑉) → ([(𝐹x) / y]ψ[B / y]ψ))
24 ralrnmpt.2 . . . . . . 7 (y = B → (ψχ))
2524sbcieg 2789 . . . . . 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 2377 . . 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 1356 . . . . 5 (x(x A → ([(𝐹x) / y]ψχ)) ↔ x((x A [(𝐹x) / y]ψ) ↔ (x A χ)))
31 exbi 1492 . . . . 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 2305 . . . 4 (x A ([(𝐹x) / y]ψχ) ↔ x(x A → ([(𝐹x) / y]ψχ)))
34 df-rex 2306 . . . . 5 (x A [(𝐹x) / y]ψx(x A [(𝐹x) / y]ψ))
35 df-rex 2306 . . . . 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 1240   = wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301  [wsbc 2758  cmpt 3809  ran crn 4289   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by: (None)
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