MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.49c Structured version   Visualization version   GIF version

Theorem tz7.49c 7933
Description: Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)
Hypothesis
Ref Expression
tz7.49c.1 𝐹 Fn On
Assertion
Ref Expression
tz7.49c ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem tz7.49c
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 tz7.49c.1 . . 3 𝐹 Fn On
2 biid 262 . . 3 (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
31, 2tz7.49 7932 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
4 3simpc 1143 . . . 4 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → ((𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
5 onss 7361 . . . . . . . . 9 (𝑥 ∈ On → 𝑥 ⊆ On)
6 fnssres 6340 . . . . . . . . 9 ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹𝑥) Fn 𝑥)
71, 5, 6sylancr 587 . . . . . . . 8 (𝑥 ∈ On → (𝐹𝑥) Fn 𝑥)
8 df-ima 5456 . . . . . . . . . 10 (𝐹𝑥) = ran (𝐹𝑥)
98eqeq1i 2800 . . . . . . . . 9 ((𝐹𝑥) = 𝐴 ↔ ran (𝐹𝑥) = 𝐴)
109biimpi 217 . . . . . . . 8 ((𝐹𝑥) = 𝐴 → ran (𝐹𝑥) = 𝐴)
117, 10anim12i 612 . . . . . . 7 ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) → ((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴))
1211anim1i 614 . . . . . 6 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
13 dff1o2 6488 . . . . . . 7 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ ((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴))
14 3anan32 1090 . . . . . . 7 (((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴) ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1513, 14bitri 276 . . . . . 6 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1612, 15sylibr 235 . . . . 5 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴)
1716expl 458 . . . 4 (𝑥 ∈ On → (((𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴))
184, 17syl5 34 . . 3 (𝑥 ∈ On → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴))
1918reximia 3206 . 2 (∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
203, 19syl 17 1 ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  cdif 3856  wss 3859  c0 4211  ccnv 5442  ran crn 5444  cres 5445  cima 5446  Oncon0 6066  Fun wfun 6219   Fn wfn 6220  1-1-ontowf1o 6224  cfv 6225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233
This theorem is referenced by:  dfac8alem  9301  dnnumch1  39129
  Copyright terms: Public domain W3C validator