Theorem isorel 7331

Description: An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)

Assertion

Ref	Expression
isorel	⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))

Proof of Theorem isorel

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-isom 6546	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
2	1	simprbi 503	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
3		breq1 5110	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑥𝑅𝑦 ↔ 𝐶𝑅𝑦))
4		fveq2 6882	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐻‘𝑥) = (𝐻‘𝐶))
5	4	breq1d 5117	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)))
6	3, 5	bibi12d 348	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦))))
7		breq2 5111	. . . 4 ⊢ (𝑦 = 𝐷 → (𝐶𝑅𝑦 ↔ 𝐶𝑅𝐷))
8		fveq2 6882	. . . . 5 ⊢ (𝑦 = 𝐷 → (𝐻‘𝑦) = (𝐻‘𝐷))
9	8	breq2d 5119	. . . 4 ⊢ (𝑦 = 𝐷 → ((𝐻‘𝐶)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
10	7, 9	bibi12d 348	. . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))))
11	6, 10	rspc2v 3590	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))))
12	2, 11	mpan9 516	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   class class class wbr 5107  –1-1-onto→wf1o 6536  ‘cfv 6537   Isom wiso 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-isom 6546

This theorem is used by:  soisores  7332  isomin  7342  isoini  7343  isopolem  7350  isosolem  7352  weniso  7361  smoiso  8355  supisolem  9448  ordiso2  9491  cantnflt  9655  cantnfp1lem3  9663  cantnflem1b  9669  cantnflem1  9672  wemapwe  9680  cnfcomlem  9682  cnfcom  9683  cnfcom3lem  9686  fpwwe2lem5  10648  fpwwe2lem6  10649  fpwwe2lem8  10651  leisorel  14529  seqcoll  14533  seqcoll2  14534  isercoll  15759  ordthmeolem  24033  iccpnfhmeo  25179  xrhmeo  25180  dvcnvrelem1  26251  dvcvx  26254  isoun  33182  erdszelem8  35785  erdsze2lem2  35791  cantnfresb  44173  fourierdlem20  46963  fourierdlem46  46988  fourierdlem50  46992  fourierdlem63  47005  fourierdlem64  47006  fourierdlem65  47007  fourierdlem76  47018  fourierdlem79  47021  fourierdlem102  47044  fourierdlem103  47045  fourierdlem104  47046  fourierdlem114  47056