MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resixpfo Structured version   Visualization version   GIF version

Theorem resixpfo 7898
Description: Restriction of elements of an infinite Cartesian product creates a surjection, if the original Cartesian product is nonempty. (Contributed by Mario Carneiro, 27-Aug-2015.)
Hypothesis
Ref Expression
resixpfo.1 𝐹 = (𝑓X𝑥𝐴 𝐶 ↦ (𝑓𝐵))
Assertion
Ref Expression
resixpfo ((𝐵𝐴X𝑥𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥𝐴 𝐶ontoX𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝑓,𝐴   𝐵,𝑓,𝑥   𝐶,𝑓
Allowed substitution hints:   𝐶(𝑥)   𝐹(𝑥,𝑓)

Proof of Theorem resixpfo
Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resixp 7895 . . . 4 ((𝐵𝐴𝑓X𝑥𝐴 𝐶) → (𝑓𝐵) ∈ X𝑥𝐵 𝐶)
2 resixpfo.1 . . . 4 𝐹 = (𝑓X𝑥𝐴 𝐶 ↦ (𝑓𝐵))
31, 2fmptd 6346 . . 3 (𝐵𝐴𝐹:X𝑥𝐴 𝐶X𝑥𝐵 𝐶)
43adantr 481 . 2 ((𝐵𝐴X𝑥𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥𝐴 𝐶X𝑥𝐵 𝐶)
5 n0 3912 . . . 4 (X𝑥𝐴 𝐶 ≠ ∅ ↔ ∃𝑔 𝑔X𝑥𝐴 𝐶)
6 eleq1 2686 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
76ifbid 4085 . . . . . . . . . . 11 (𝑧 = 𝑥 → if(𝑧𝐵, , 𝑔) = if(𝑥𝐵, , 𝑔))
8 id 22 . . . . . . . . . . 11 (𝑧 = 𝑥𝑧 = 𝑥)
97, 8fveq12d 6159 . . . . . . . . . 10 (𝑧 = 𝑥 → (if(𝑧𝐵, , 𝑔)‘𝑧) = (if(𝑥𝐵, , 𝑔)‘𝑥))
109cbvmptv 4715 . . . . . . . . 9 (𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) = (𝑥𝐴 ↦ (if(𝑥𝐵, , 𝑔)‘𝑥))
11 vex 3192 . . . . . . . . . . . . 13 𝑔 ∈ V
1211elixp 7867 . . . . . . . . . . . 12 (𝑔X𝑥𝐴 𝐶 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐶))
1312simprbi 480 . . . . . . . . . . 11 (𝑔X𝑥𝐴 𝐶 → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐶)
14 vex 3192 . . . . . . . . . . . . . . . . 17 ∈ V
1514elixp 7867 . . . . . . . . . . . . . . . 16 (X𝑥𝐵 𝐶 ↔ ( Fn 𝐵 ∧ ∀𝑥𝐵 (𝑥) ∈ 𝐶))
1615simprbi 480 . . . . . . . . . . . . . . 15 (X𝑥𝐵 𝐶 → ∀𝑥𝐵 (𝑥) ∈ 𝐶)
17 fveq1 6152 . . . . . . . . . . . . . . . . . . 19 ( = if(𝑥𝐵, , 𝑔) → (𝑥) = (if(𝑥𝐵, , 𝑔)‘𝑥))
1817eleq1d 2683 . . . . . . . . . . . . . . . . . 18 ( = if(𝑥𝐵, , 𝑔) → ((𝑥) ∈ 𝐶 ↔ (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶))
19 fveq1 6152 . . . . . . . . . . . . . . . . . . 19 (𝑔 = if(𝑥𝐵, , 𝑔) → (𝑔𝑥) = (if(𝑥𝐵, , 𝑔)‘𝑥))
2019eleq1d 2683 . . . . . . . . . . . . . . . . . 18 (𝑔 = if(𝑥𝐵, , 𝑔) → ((𝑔𝑥) ∈ 𝐶 ↔ (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶))
21 simpl 473 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝐵 → (𝑥) ∈ 𝐶) ∧ (𝑥𝐴 ∧ (𝑔𝑥) ∈ 𝐶)) → (𝑥𝐵 → (𝑥) ∈ 𝐶))
2221imp 445 . . . . . . . . . . . . . . . . . 18 ((((𝑥𝐵 → (𝑥) ∈ 𝐶) ∧ (𝑥𝐴 ∧ (𝑔𝑥) ∈ 𝐶)) ∧ 𝑥𝐵) → (𝑥) ∈ 𝐶)
23 simplrr 800 . . . . . . . . . . . . . . . . . 18 ((((𝑥𝐵 → (𝑥) ∈ 𝐶) ∧ (𝑥𝐴 ∧ (𝑔𝑥) ∈ 𝐶)) ∧ ¬ 𝑥𝐵) → (𝑔𝑥) ∈ 𝐶)
2418, 20, 22, 23ifbothda 4100 . . . . . . . . . . . . . . . . 17 (((𝑥𝐵 → (𝑥) ∈ 𝐶) ∧ (𝑥𝐴 ∧ (𝑔𝑥) ∈ 𝐶)) → (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶)
2524exp32 630 . . . . . . . . . . . . . . . 16 ((𝑥𝐵 → (𝑥) ∈ 𝐶) → (𝑥𝐴 → ((𝑔𝑥) ∈ 𝐶 → (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶)))
2625ralimi2 2944 . . . . . . . . . . . . . . 15 (∀𝑥𝐵 (𝑥) ∈ 𝐶 → ∀𝑥𝐴 ((𝑔𝑥) ∈ 𝐶 → (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶))
2716, 26syl 17 . . . . . . . . . . . . . 14 (X𝑥𝐵 𝐶 → ∀𝑥𝐴 ((𝑔𝑥) ∈ 𝐶 → (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶))
2827adantl 482 . . . . . . . . . . . . 13 ((𝐵𝐴X𝑥𝐵 𝐶) → ∀𝑥𝐴 ((𝑔𝑥) ∈ 𝐶 → (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶))
29 ralim 2943 . . . . . . . . . . . . 13 (∀𝑥𝐴 ((𝑔𝑥) ∈ 𝐶 → (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶) → (∀𝑥𝐴 (𝑔𝑥) ∈ 𝐶 → ∀𝑥𝐴 (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶))
3028, 29syl 17 . . . . . . . . . . . 12 ((𝐵𝐴X𝑥𝐵 𝐶) → (∀𝑥𝐴 (𝑔𝑥) ∈ 𝐶 → ∀𝑥𝐴 (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶))
3130imp 445 . . . . . . . . . . 11 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐶) → ∀𝑥𝐴 (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶)
3213, 31sylan2 491 . . . . . . . . . 10 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → ∀𝑥𝐴 (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶)
33 n0i 3901 . . . . . . . . . . . . 13 (𝑔X𝑥𝐴 𝐶 → ¬ X𝑥𝐴 𝐶 = ∅)
34 ixpprc 7881 . . . . . . . . . . . . 13 𝐴 ∈ V → X𝑥𝐴 𝐶 = ∅)
3533, 34nsyl2 142 . . . . . . . . . . . 12 (𝑔X𝑥𝐴 𝐶𝐴 ∈ V)
3635adantl 482 . . . . . . . . . . 11 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → 𝐴 ∈ V)
37 mptelixpg 7897 . . . . . . . . . . 11 (𝐴 ∈ V → ((𝑥𝐴 ↦ (if(𝑥𝐵, , 𝑔)‘𝑥)) ∈ X𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶))
3836, 37syl 17 . . . . . . . . . 10 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → ((𝑥𝐴 ↦ (if(𝑥𝐵, , 𝑔)‘𝑥)) ∈ X𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 (if(𝑥𝐵, , 𝑔)‘𝑥) ∈ 𝐶))
3932, 38mpbird 247 . . . . . . . . 9 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → (𝑥𝐴 ↦ (if(𝑥𝐵, , 𝑔)‘𝑥)) ∈ X𝑥𝐴 𝐶)
4010, 39syl5eqel 2702 . . . . . . . 8 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → (𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ∈ X𝑥𝐴 𝐶)
41 iftrue 4069 . . . . . . . . . . . . . 14 (𝑧𝐵 → if(𝑧𝐵, , 𝑔) = )
4241fveq1d 6155 . . . . . . . . . . . . 13 (𝑧𝐵 → (if(𝑧𝐵, , 𝑔)‘𝑧) = (𝑧))
4342mpteq2ia 4705 . . . . . . . . . . . 12 (𝑧𝐵 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) = (𝑧𝐵 ↦ (𝑧))
44 resmpt 5413 . . . . . . . . . . . . 13 (𝐵𝐴 → ((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ↾ 𝐵) = (𝑧𝐵 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)))
4544ad2antrr 761 . . . . . . . . . . . 12 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → ((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ↾ 𝐵) = (𝑧𝐵 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)))
46 ixpfn 7866 . . . . . . . . . . . . . 14 (X𝑥𝐵 𝐶 Fn 𝐵)
4746ad2antlr 762 . . . . . . . . . . . . 13 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → Fn 𝐵)
48 dffn5 6203 . . . . . . . . . . . . 13 ( Fn 𝐵 = (𝑧𝐵 ↦ (𝑧)))
4947, 48sylib 208 . . . . . . . . . . . 12 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → = (𝑧𝐵 ↦ (𝑧)))
5043, 45, 493eqtr4a 2681 . . . . . . . . . . 11 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → ((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ↾ 𝐵) = )
5150, 14syl6eqel 2706 . . . . . . . . . 10 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → ((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ↾ 𝐵) ∈ V)
52 reseq1 5355 . . . . . . . . . . 11 (𝑓 = (𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) → (𝑓𝐵) = ((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ↾ 𝐵))
5352, 2fvmptg 6242 . . . . . . . . . 10 (((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ∈ X𝑥𝐴 𝐶 ∧ ((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ↾ 𝐵) ∈ V) → (𝐹‘(𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧))) = ((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ↾ 𝐵))
5440, 51, 53syl2anc 692 . . . . . . . . 9 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → (𝐹‘(𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧))) = ((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ↾ 𝐵))
5554, 50eqtr2d 2656 . . . . . . . 8 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → = (𝐹‘(𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧))))
56 fveq2 6153 . . . . . . . . . 10 (𝑦 = (𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) → (𝐹𝑦) = (𝐹‘(𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧))))
5756eqeq2d 2631 . . . . . . . . 9 (𝑦 = (𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) → ( = (𝐹𝑦) ↔ = (𝐹‘(𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)))))
5857rspcev 3298 . . . . . . . 8 (((𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)) ∈ X𝑥𝐴 𝐶 = (𝐹‘(𝑧𝐴 ↦ (if(𝑧𝐵, , 𝑔)‘𝑧)))) → ∃𝑦X 𝑥𝐴 𝐶 = (𝐹𝑦))
5940, 55, 58syl2anc 692 . . . . . . 7 (((𝐵𝐴X𝑥𝐵 𝐶) ∧ 𝑔X𝑥𝐴 𝐶) → ∃𝑦X 𝑥𝐴 𝐶 = (𝐹𝑦))
6059ex 450 . . . . . 6 ((𝐵𝐴X𝑥𝐵 𝐶) → (𝑔X𝑥𝐴 𝐶 → ∃𝑦X 𝑥𝐴 𝐶 = (𝐹𝑦)))
6160ralrimdva 2964 . . . . 5 (𝐵𝐴 → (𝑔X𝑥𝐴 𝐶 → ∀X 𝑥𝐵 𝐶𝑦X 𝑥𝐴 𝐶 = (𝐹𝑦)))
6261exlimdv 1858 . . . 4 (𝐵𝐴 → (∃𝑔 𝑔X𝑥𝐴 𝐶 → ∀X 𝑥𝐵 𝐶𝑦X 𝑥𝐴 𝐶 = (𝐹𝑦)))
635, 62syl5bi 232 . . 3 (𝐵𝐴 → (X𝑥𝐴 𝐶 ≠ ∅ → ∀X 𝑥𝐵 𝐶𝑦X 𝑥𝐴 𝐶 = (𝐹𝑦)))
6463imp 445 . 2 ((𝐵𝐴X𝑥𝐴 𝐶 ≠ ∅) → ∀X 𝑥𝐵 𝐶𝑦X 𝑥𝐴 𝐶 = (𝐹𝑦))
65 dffo3 6335 . 2 (𝐹:X𝑥𝐴 𝐶ontoX𝑥𝐵 𝐶 ↔ (𝐹:X𝑥𝐴 𝐶X𝑥𝐵 𝐶 ∧ ∀X 𝑥𝐵 𝐶𝑦X 𝑥𝐴 𝐶 = (𝐹𝑦)))
664, 64, 65sylanbrc 697 1 ((𝐵𝐴X𝑥𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥𝐴 𝐶ontoX𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3189  wss 3559  c0 3896  ifcif 4063  cmpt 4678  cres 5081   Fn wfn 5847  wf 5848  ontowfo 5850  cfv 5852  Xcixp 7860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ixp 7861
This theorem is referenced by:  ptcmplem2  21780
  Copyright terms: Public domain W3C validator