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Theorem ixpssmapc 39557
Description: An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
ixpssmapc.x 𝑥𝜑
ixpssmapc.c (𝜑𝐶𝑉)
ixpssmapc.b ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
ixpssmapc (𝜑X𝑥𝐴 𝐵 ⊆ (𝐶𝑚 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem ixpssmapc
StepHypRef Expression
1 ixpssmapc.c . . . 4 (𝜑𝐶𝑉)
2 ixpssmapc.x . . . . . 6 𝑥𝜑
3 ixpssmapc.b . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝐶)
43ex 449 . . . . . 6 (𝜑 → (𝑥𝐴𝐵𝐶))
52, 4ralrimi 2986 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
6 iunss 4593 . . . . 5 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
75, 6sylibr 224 . . . 4 (𝜑 𝑥𝐴 𝐵𝐶)
81, 7ssexd 4838 . . 3 (𝜑 𝑥𝐴 𝐵 ∈ V)
9 ixpssmap2g 7979 . . 3 ( 𝑥𝐴 𝐵 ∈ V → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
108, 9syl 17 . 2 (𝜑X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
11 mapss 7942 . . 3 ((𝐶𝑉 𝑥𝐴 𝐵𝐶) → ( 𝑥𝐴 𝐵𝑚 𝐴) ⊆ (𝐶𝑚 𝐴))
121, 7, 11syl2anc 694 . 2 (𝜑 → ( 𝑥𝐴 𝐵𝑚 𝐴) ⊆ (𝐶𝑚 𝐴))
1310, 12sstrd 3646 1 (𝜑X𝑥𝐴 𝐵 ⊆ (𝐶𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wnf 1748  wcel 2030  wral 2941  Vcvv 3231  wss 3607   ciun 4552  (class class class)co 6690  𝑚 cmap 7899  Xcixp 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-ixp 7951
This theorem is referenced by:  ioorrnopnlem  40842
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