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Theorem dflec3 6221
 Description: Another potential definition of cardinal inequality. (Contributed by SF, 23-Mar-2015.)
Assertion
Ref Expression
dflec3 ((M NC N NC ) → (Mc Na M b N f f:a1-1b))
Distinct variable groups:   M,a   N,a,b   f,a,b
Allowed substitution hints:   M(f,b)   N(f)

Proof of Theorem dflec3
Dummy variables x y c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elncs 6119 . . . 4 (M NCx M = Nc x)
2 elncs 6119 . . . 4 (N NCy N = Nc y)
31, 2anbi12i 678 . . 3 ((M NC N NC ) ↔ (x M = Nc x y N = Nc y))
4 eeanv 1913 . . 3 (xy(M = Nc x N = Nc y) ↔ (x M = Nc x y N = Nc y))
53, 4bitr4i 243 . 2 ((M NC N NC ) ↔ xy(M = Nc x N = Nc y))
6 ncex 6117 . . . . . 6 Nc x V
7 ncex 6117 . . . . . 6 Nc y V
86, 7brlec 6113 . . . . 5 ( Nc xc Nc yc Nc xb Nc yc b)
9 rexcom 2772 . . . . . 6 (c Nc xb Nc yc bb Nc yc Nc xc b)
10 f1oi 5320 . . . . . . . . . . . . 13 ( I c):c1-1-ontoc
11 f1of1 5286 . . . . . . . . . . . . 13 (( I c):c1-1-ontoc → ( I c):c1-1c)
1210, 11ax-mp 8 . . . . . . . . . . . 12 ( I c):c1-1c
13 f1ss 5262 . . . . . . . . . . . 12 ((( I c):c1-1c c b) → ( I c):c1-1b)
1412, 13mpan 651 . . . . . . . . . . 11 (c b → ( I c):c1-1b)
15 idex 5504 . . . . . . . . . . . . 13 I V
16 vex 2862 . . . . . . . . . . . . 13 c V
1715, 16resex 5117 . . . . . . . . . . . 12 ( I c) V
18 f1eq1 5253 . . . . . . . . . . . 12 (f = ( I c) → (f:c1-1b ↔ ( I c):c1-1b))
1917, 18spcev 2946 . . . . . . . . . . 11 (( I c):c1-1bf f:c1-1b)
2014, 19syl 15 . . . . . . . . . 10 (c bf f:c1-1b)
21 f1eq2 5254 . . . . . . . . . . . 12 (a = c → (f:a1-1bf:c1-1b))
2221exbidv 1626 . . . . . . . . . . 11 (a = c → (f f:a1-1bf f:c1-1b))
2322rspcev 2955 . . . . . . . . . 10 ((c Nc x f f:c1-1b) → a Nc xf f:a1-1b)
2420, 23sylan2 460 . . . . . . . . 9 ((c Nc x c b) → a Nc xf f:a1-1b)
2524rexlimiva 2733 . . . . . . . 8 (c Nc xc ba Nc xf f:a1-1b)
26 vex 2862 . . . . . . . . . . . . . . . 16 a V
2726eqnc 6127 . . . . . . . . . . . . . . 15 ( Nc a = Nc xax)
28 elnc 6125 . . . . . . . . . . . . . . 15 (a Nc xax)
2927, 28bitr4i 243 . . . . . . . . . . . . . 14 ( Nc a = Nc xa Nc x)
30 f1f1orn 5297 . . . . . . . . . . . . . . . . . 18 (f:a1-1bf:a1-1-onto→ran f)
31 vex 2862 . . . . . . . . . . . . . . . . . . 19 f V
3231f1oen 6033 . . . . . . . . . . . . . . . . . 18 (f:a1-1-onto→ran fa ≈ ran f)
3330, 32syl 15 . . . . . . . . . . . . . . . . 17 (f:a1-1ba ≈ ran f)
34 ensym 6037 . . . . . . . . . . . . . . . . 17 (a ≈ ran f ↔ ran fa)
3533, 34sylib 188 . . . . . . . . . . . . . . . 16 (f:a1-1b → ran fa)
36 elnc 6125 . . . . . . . . . . . . . . . 16 (ran f Nc a ↔ ran fa)
3735, 36sylibr 203 . . . . . . . . . . . . . . 15 (f:a1-1b → ran f Nc a)
38 eleq2 2414 . . . . . . . . . . . . . . 15 ( Nc a = Nc x → (ran f Nc a ↔ ran f Nc x))
3937, 38syl5ib 210 . . . . . . . . . . . . . 14 ( Nc a = Nc x → (f:a1-1b → ran f Nc x))
4029, 39sylbir 204 . . . . . . . . . . . . 13 (a Nc x → (f:a1-1b → ran f Nc x))
4140imp 418 . . . . . . . . . . . 12 ((a Nc x f:a1-1b) → ran f Nc x)
42 f1f 5258 . . . . . . . . . . . . . 14 (f:a1-1bf:a–→b)
43 frn 5228 . . . . . . . . . . . . . 14 (f:a–→b → ran f b)
4442, 43syl 15 . . . . . . . . . . . . 13 (f:a1-1b → ran f b)
4544adantl 452 . . . . . . . . . . . 12 ((a Nc x f:a1-1b) → ran f b)
46 sseq1 3292 . . . . . . . . . . . . 13 (c = ran f → (c b ↔ ran f b))
4746rspcev 2955 . . . . . . . . . . . 12 ((ran f Nc x ran f b) → c Nc xc b)
4841, 45, 47syl2anc 642 . . . . . . . . . . 11 ((a Nc x f:a1-1b) → c Nc xc b)
4948ex 423 . . . . . . . . . 10 (a Nc x → (f:a1-1bc Nc xc b))
5049exlimdv 1636 . . . . . . . . 9 (a Nc x → (f f:a1-1bc Nc xc b))
5150rexlimiv 2732 . . . . . . . 8 (a Nc xf f:a1-1bc Nc xc b)
5225, 51impbii 180 . . . . . . 7 (c Nc xc ba Nc xf f:a1-1b)
5352rexbii 2639 . . . . . 6 (b Nc yc Nc xc bb Nc ya Nc xf f:a1-1b)
549, 53bitri 240 . . . . 5 (c Nc xb Nc yc bb Nc ya Nc xf f:a1-1b)
55 rexcom 2772 . . . . 5 (b Nc ya Nc xf f:a1-1ba Nc xb Nc yf f:a1-1b)
568, 54, 553bitri 262 . . . 4 ( Nc xc Nc ya Nc xb Nc yf f:a1-1b)
57 breq12 4644 . . . . 5 ((M = Nc x N = Nc y) → (Mc NNc xc Nc y))
58 simpl 443 . . . . . 6 ((M = Nc x N = Nc y) → M = Nc x)
59 rexeq 2808 . . . . . . 7 (N = Nc y → (b N f f:a1-1bb Nc yf f:a1-1b))
6059adantl 452 . . . . . 6 ((M = Nc x N = Nc y) → (b N f f:a1-1bb Nc yf f:a1-1b))
6158, 60rexeqbidv 2820 . . . . 5 ((M = Nc x N = Nc y) → (a M b N f f:a1-1ba Nc xb Nc yf f:a1-1b))
6257, 61bibi12d 312 . . . 4 ((M = Nc x N = Nc y) → ((Mc Na M b N f f:a1-1b) ↔ ( Nc xc Nc ya Nc xb Nc yf f:a1-1b)))
6356, 62mpbiri 224 . . 3 ((M = Nc x N = Nc y) → (Mc Na M b N f f:a1-1b))
6463exlimivv 1635 . 2 (xy(M = Nc x N = Nc y) → (Mc Na M b N f f:a1-1b))
655, 64sylbi 187 1 ((M NC N NC ) → (Mc Na M b N f f:a1-1b))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ⊆ wss 3257   class class class wbr 4639   I cid 4763  ran crn 4773   ↾ cres 4774  –→wf 4777  –1-1→wf1 4778  –1-1-onto→wf1o 4780   ≈ cen 6028   NC cncs 6088   ≤c clec 6089   Nc cnc 6091 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101 This theorem is referenced by:  nclenc  6222
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