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Theorem tsrlemax 18541
Description: Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
istsr.1 𝑋 = dom 𝑅
Assertion
Ref Expression
tsrlemax ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))

Proof of Theorem tsrlemax
StepHypRef Expression
1 breq2 5152 . . 3 (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐶𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
21bibi1d 343 . 2 (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶))))
3 breq2 5152 . . 3 (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐵𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
43bibi1d 343 . 2 (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶))))
5 olc 866 . . 3 (𝐴𝑅𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
6 eqid 2732 . . . . . . . . . 10 dom 𝑅 = dom 𝑅
76istsr 18538 . . . . . . . . 9 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
87simplbi 498 . . . . . . . 8 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
9 pstr 18532 . . . . . . . . 9 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1093expib 1122 . . . . . . . 8 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
118, 10syl 17 . . . . . . 7 (𝑅 ∈ TosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
1211adantr 481 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
1312expdimp 453 . . . . 5 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐶𝐴𝑅𝐶))
1413impancom 452 . . . 4 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶))
15 idd 24 . . . 4 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶𝐴𝑅𝐶))
1614, 15jaod 857 . . 3 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵𝐴𝑅𝐶) → 𝐴𝑅𝐶))
175, 16impbid2 225 . 2 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
18 orc 865 . . 3 (𝐴𝑅𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶))
19 idd 24 . . . 4 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐵))
20 istsr.1 . . . . . . . 8 𝑋 = dom 𝑅
2120tsrlin 18540 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝑅𝐶𝐶𝑅𝐵))
22213adant3r1 1182 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝑅𝐶𝐶𝑅𝐵))
2322orcanai 1001 . . . . 5 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → 𝐶𝑅𝐵)
24 pstr 18532 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐶𝐶𝑅𝐵) → 𝐴𝑅𝐵)
25243expib 1122 . . . . . . . . 9 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐶𝐶𝑅𝐵) → 𝐴𝑅𝐵))
268, 25syl 17 . . . . . . . 8 (𝑅 ∈ TosetRel → ((𝐴𝑅𝐶𝐶𝑅𝐵) → 𝐴𝑅𝐵))
2726adantr 481 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐶𝐶𝑅𝐵) → 𝐴𝑅𝐵))
2827expdimp 453 . . . . . 6 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅𝐶) → (𝐶𝑅𝐵𝐴𝑅𝐵))
2928impancom 452 . . . . 5 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑅𝐵) → (𝐴𝑅𝐶𝐴𝑅𝐵))
3023, 29syldan 591 . . . 4 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐶𝐴𝑅𝐵))
3119, 30jaod 857 . . 3 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵𝐴𝑅𝐶) → 𝐴𝑅𝐵))
3218, 31impbid2 225 . 2 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
332, 4, 17, 32ifbothda 4566 1 ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  cun 3946  wss 3948  ifcif 4528   class class class wbr 5148   × cxp 5674  ccnv 5675  dom cdm 5676  PosetRelcps 18519   TosetRel ctsr 18520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-ps 18521  df-tsr 18522
This theorem is referenced by:  ordtbaslem  22699
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