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Theorem ixpssmapg 8467
Description: An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)
Assertion
Ref Expression
ixpssmapg (∀𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem ixpssmapg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 n0i 4272 . . . . . . 7 (𝑓X𝑥𝐴 𝐵 → ¬ X𝑥𝐴 𝐵 = ∅)
2 ixpprc 8458 . . . . . . 7 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
31, 2nsyl2 143 . . . . . 6 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
4 id 22 . . . . . 6 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵𝑉)
5 iunexg 7639 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑥𝐴 𝐵𝑉) → 𝑥𝐴 𝐵 ∈ V)
63, 4, 5syl2anr 599 . . . . 5 ((∀𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑥𝐴 𝐵 ∈ V)
7 ixpssmap2g 8466 . . . . 5 ( 𝑥𝐴 𝐵 ∈ V → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
86, 7syl 17 . . . 4 ((∀𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
9 simpr 488 . . . 4 ((∀𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓X𝑥𝐴 𝐵)
108, 9sseldd 3944 . . 3 ((∀𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴))
1110ex 416 . 2 (∀𝑥𝐴 𝐵𝑉 → (𝑓X𝑥𝐴 𝐵𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴)))
1211ssrdv 3949 1 (∀𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3126  Vcvv 3471  wss 3910  c0 4266   ciun 4892  (class class class)co 7130  m cmap 8381  Xcixp 8436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-ixp 8437
This theorem is referenced by:  ixpssmap  8471  gruixp  10208  hoissrrn  42981  hoissrrn2  43010
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