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Theorem sizeusglecusglem1 26012
 Description: Lemma 1 for sizeusglecusg 26014. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
Assertion
Ref Expression
sizeusglecusglem1 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐹)

Proof of Theorem sizeusglecusglem1
Dummy variables 𝑎 𝑏 𝑒 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 6086 . . 3 ( I ↾ ran 𝐸):ran 𝐸1-1-onto→ran 𝐸
2 f1of1 6049 . . 3 (( I ↾ ran 𝐸):ran 𝐸1-1-onto→ran 𝐸 → ( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐸)
31, 2ax-mp 5 . 2 ( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐸
4 iscusgra0 25986 . . . . 5 (𝑉 ComplUSGrph 𝐹 → (𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹))
5 usgrarnedg 25913 . . . . 5 ((𝑉 USGrph 𝐸𝑒 ∈ ran 𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑒 = {𝑎, 𝑏}))
6 simprr 792 . . . . . . . . . . . . . . 15 ((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) → 𝑏𝑉)
7 eldifsn 4260 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ (𝑉 ∖ {𝑏}) ↔ (𝑎𝑉𝑎𝑏))
87simplbi2 653 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑉 → (𝑎𝑏𝑎 ∈ (𝑉 ∖ {𝑏})))
98adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑉𝑏𝑉) → (𝑎𝑏𝑎 ∈ (𝑉 ∖ {𝑏})))
109impcom 445 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) → 𝑎 ∈ (𝑉 ∖ {𝑏}))
1110adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑥 = 𝑏) → 𝑎 ∈ (𝑉 ∖ {𝑏}))
12 sneq 4135 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑏 → {𝑥} = {𝑏})
1312difeq2d 3690 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑏 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑏}))
1413eleq2d 2673 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑏 → (𝑎 ∈ (𝑉 ∖ {𝑥}) ↔ 𝑎 ∈ (𝑉 ∖ {𝑏})))
1514adantl 481 . . . . . . . . . . . . . . . . 17 (((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑥 = 𝑏) → (𝑎 ∈ (𝑉 ∖ {𝑥}) ↔ 𝑎 ∈ (𝑉 ∖ {𝑏})))
1611, 15mpbird 246 . . . . . . . . . . . . . . . 16 (((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑥 = 𝑏) → 𝑎 ∈ (𝑉 ∖ {𝑥}))
17 preq12 4214 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 = 𝑎𝑥 = 𝑏) → {𝑦, 𝑥} = {𝑎, 𝑏})
1817expcom 450 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑏 → (𝑦 = 𝑎 → {𝑦, 𝑥} = {𝑎, 𝑏}))
1918adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑥 = 𝑏) → (𝑦 = 𝑎 → {𝑦, 𝑥} = {𝑎, 𝑏}))
2019imp 444 . . . . . . . . . . . . . . . . 17 ((((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑥 = 𝑏) ∧ 𝑦 = 𝑎) → {𝑦, 𝑥} = {𝑎, 𝑏})
2120eleq1d 2672 . . . . . . . . . . . . . . . 16 ((((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑥 = 𝑏) ∧ 𝑦 = 𝑎) → ({𝑦, 𝑥} ∈ ran 𝐹 ↔ {𝑎, 𝑏} ∈ ran 𝐹))
2216, 21rspcdv 3285 . . . . . . . . . . . . . . 15 (((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑥 = 𝑏) → (∀𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹 → {𝑎, 𝑏} ∈ ran 𝐹))
236, 22rspcimdv 3283 . . . . . . . . . . . . . 14 ((𝑎𝑏 ∧ (𝑎𝑉𝑏𝑉)) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹 → {𝑎, 𝑏} ∈ ran 𝐹))
2423ex 449 . . . . . . . . . . . . 13 (𝑎𝑏 → ((𝑎𝑉𝑏𝑉) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹 → {𝑎, 𝑏} ∈ ran 𝐹)))
2524com13 86 . . . . . . . . . . . 12 (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹 → ((𝑎𝑉𝑏𝑉) → (𝑎𝑏 → {𝑎, 𝑏} ∈ ran 𝐹)))
2625adantl 481 . . . . . . . . . . 11 ((𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹) → ((𝑎𝑉𝑏𝑉) → (𝑎𝑏 → {𝑎, 𝑏} ∈ ran 𝐹)))
2726imp 444 . . . . . . . . . 10 (((𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹) ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → {𝑎, 𝑏} ∈ ran 𝐹))
2827com12 32 . . . . . . . . 9 (𝑎𝑏 → (((𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹) ∧ (𝑎𝑉𝑏𝑉)) → {𝑎, 𝑏} ∈ ran 𝐹))
29 eleq1 2676 . . . . . . . . . 10 (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ ran 𝐹 ↔ {𝑎, 𝑏} ∈ ran 𝐹))
3029imbi2d 329 . . . . . . . . 9 (𝑒 = {𝑎, 𝑏} → ((((𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹) ∧ (𝑎𝑉𝑏𝑉)) → 𝑒 ∈ ran 𝐹) ↔ (((𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹) ∧ (𝑎𝑉𝑏𝑉)) → {𝑎, 𝑏} ∈ ran 𝐹)))
3128, 30syl5ibr 235 . . . . . . . 8 (𝑒 = {𝑎, 𝑏} → (𝑎𝑏 → (((𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹) ∧ (𝑎𝑉𝑏𝑉)) → 𝑒 ∈ ran 𝐹)))
3231impcom 445 . . . . . . 7 ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → (((𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹) ∧ (𝑎𝑉𝑏𝑉)) → 𝑒 ∈ ran 𝐹))
3332com12 32 . . . . . 6 (((𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹) ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → 𝑒 ∈ ran 𝐹))
3433rexlimdvva 3020 . . . . 5 ((𝑉 USGrph 𝐹 ∧ ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}){𝑦, 𝑥} ∈ ran 𝐹) → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑒 = {𝑎, 𝑏}) → 𝑒 ∈ ran 𝐹))
354, 5, 34syl2imc 40 . . . 4 ((𝑉 USGrph 𝐸𝑒 ∈ ran 𝐸) → (𝑉 ComplUSGrph 𝐹𝑒 ∈ ran 𝐹))
3635impancom 455 . . 3 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (𝑒 ∈ ran 𝐸𝑒 ∈ ran 𝐹))
3736ssrdv 3574 . 2 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ran 𝐸 ⊆ ran 𝐹)
38 f1ss 6019 . 2 ((( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐸 ∧ ran 𝐸 ⊆ ran 𝐹) → ( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐹)
393, 37, 38sylancr 694 1 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583   I cid 4948  ran crn 5039   ↾ cres 5040  –1-1→wf1 5801  –1-1-onto→wf1o 5803   USGrph cusg 25859   ComplUSGrph ccusgra 25947 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-cusgra 25950 This theorem is referenced by:  sizeusglecusg  26014
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