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 7711
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 251 . . 3 (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
31, 2tz7.49 7710 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
4 3simpc 1147 . . . 4 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → ((𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
5 onss 7156 . . . . . . . . 9 (𝑥 ∈ On → 𝑥 ⊆ On)
6 fnssres 6165 . . . . . . . . 9 ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹𝑥) Fn 𝑥)
71, 5, 6sylancr 698 . . . . . . . 8 (𝑥 ∈ On → (𝐹𝑥) Fn 𝑥)
8 df-ima 5279 . . . . . . . . . 10 (𝐹𝑥) = ran (𝐹𝑥)
98eqeq1i 2765 . . . . . . . . 9 ((𝐹𝑥) = 𝐴 ↔ ran (𝐹𝑥) = 𝐴)
109biimpi 206 . . . . . . . 8 ((𝐹𝑥) = 𝐴 → ran (𝐹𝑥) = 𝐴)
117, 10anim12i 591 . . . . . . 7 ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) → ((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴))
1211anim1i 593 . . . . . 6 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
13 dff1o2 6304 . . . . . . 7 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ ((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴))
14 3anan32 1083 . . . . . . 7 (((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴) ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1513, 14bitri 264 . . . . . 6 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1612, 15sylibr 224 . . . . 5 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴)
1716expl 649 . . . 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 3147 . 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 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  cdif 3712  wss 3715  c0 4058  ccnv 5265  ran crn 5267  cres 5268  cima 5269  Oncon0 5884  Fun wfun 6043   Fn wfn 6044  1-1-ontowf1o 6048  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057
This theorem is referenced by:  dfac8alem  9062  dnnumch1  38134
  Copyright terms: Public domain W3C validator